(C2xQ8).36D6 - GroupNames

G = (C₂×Q₈).₃₆D₆ order 192 = 2⁶·3

12^nd non-split extension by C₂×Q₈ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄⋊C₄.₂₀D₆, (C₂×C₈).₁₇₄D₆, (C₂×Q₈).₃₆D₆, C₂₄⋊C₄.₇C₂, Q₈⋊C₄.₈S₃, C₄.₃₀(C₄○D₁₂), C₁₂.₁₆(C₄○D₄), (C₂×Dic₃).₂₇D₄, C₆.SD₁₆.₃C₂, Q₈⋊₂Dic₃.₅C₂, C₂².₁₉₁(S₃×D₄), C₄.Dic₆.₂C₂, (C₆×Q₈).₂₀C₂², C₄.₅₆(D₄⋊₂S₃), (C₂×C₂₄).₂₃₇C₂², (C₂×C₁₂).₂₃₇C₂³, Dic₃⋊Q₈.₄C₂, C₂.₁₀(Q₁₆⋊S₃), C₂.₁₆(D₄.D₆), C₆.₂₈(C₄.₄D₄), C₆.₅₅(C₈.C₂²), C₄⋊Dic₃.₈₆C₂², C₂.Dic₁₂.₁₂C₂, (C₂×Dic₆).₆₆C₂², (C₄×Dic₃).₂₁C₂², C₃⋊₂(C₄².₃₀C₂²), C₂.₁₈(C₂³.₁₁D₆), (C₂×C₆).₂₅₀(C₂×D₄), (C₂×C₃⋊C₈).₃₂C₂², (C₃×C₄⋊C₄).₃₈C₂², (C₂×C₄).₃₄₄(C₂²×S₃), (C₃×Q₈⋊C₄).₁₂C₂, SmallGroup(192,356)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — (C₂×Q₈).₃₆D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₄×Dic₃ — Dic₃⋊Q₈ — (C₂×Q₈).₃₆D₆

Lower central

C₃ — C₆ — C₂×C₁₂ — (C₂×Q₈).₃₆D₆

Upper central

C₁ — C₂² — C₂×C₄ — Q₈⋊C₄

Generators and relations for (C₂×Q₈).₃₆D₆
G = < a,b,c,d,e | a²=b⁴=1, c²=b², d⁶=ab², e²=a, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=dbd^-1=ebe^-1=b^-1, dcd^-1=ab^-1c, ece^-1=bc, ede^-1=ad⁵ >

Subgroups: 232 in 90 conjugacy classes, 37 normal (all characteristic)
C₁, C₂, C₃, C₄, C₄, C₂², C₆, C₈, C₂×C₄, C₂×C₄, Q₈, Dic₃, C₁₂, C₁₂, C₂×C₆, C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×Q₈, C₂×Q₈, C₃⋊C₈, C₂₄, Dic₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₈⋊C₄, Q₈⋊C₄, Q₈⋊C₄, C₄².C₂, C₄⋊Q₈, C₂×C₃⋊C₈, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, C₄⋊Dic₃, C₃×C₄⋊C₄, C₂×C₂₄, C₂×Dic₆, C₆×Q₈, C₄².₃₀C₂², C₆.SD₁₆, C₂₄⋊C₄, C₂.Dic₁₂, Q₈⋊₂Dic₃, C₃×Q₈⋊C₄, C₄.Dic₆, Dic₃⋊Q₈, (C₂×Q₈).₃₆D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₄○D₄, C₂²×S₃, C₄.₄D₄, C₈.C₂², C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, C₄².₃₀C₂², C₂³.₁₁D₆, D₄.D₆, Q₁₆⋊S₃, (C₂×Q₈).₃₆D₆

Character table of (C₂×Q₈).₃₆D₆

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	24A	24B	24C	24D
size	1	1	1	1	2	2	2	8	8	12	12	24	24	2	2	2	4	4	12	12	4	4	8	8	8	8	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	-1	-1	1	1	1	1	-1	1	1	-1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	-1	1	1	1	-1	1	1	1	-1	-1	-1	-1	1	1	-1	1	1	-1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	1	1	1	1	1	-1	1	1	1	-1	1	1	1	1	-1	-1	-1	-1	1	1	1	-1	-1	1	-1	-1	-1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	-1	-1	1	1	1	1	linear of order 2
ρ₇	1	1	1	1	1	1	1	-1	1	-1	-1	1	-1	1	1	1	-1	-1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	linear of order 2
ρ₉	2	2	2	2	-1	2	2	2	2	0	0	0	0	-1	-1	-1	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₀	2	2	2	2	-1	2	2	-2	-2	0	0	0	0	-1	-1	-1	2	2	0	0	-1	-1	1	1	1	1	-1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₁	2	2	2	2	2	-2	-2	0	0	-2	2	0	0	2	2	2	0	0	0	0	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	-2	0	0	2	-2	0	0	2	2	2	0	0	0	0	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-1	2	2	2	-2	0	0	0	0	-1	-1	-1	-2	-2	0	0	-1	-1	1	-1	-1	1	1	1	1	1	orthogonal lifted from D₆
ρ₁₄	2	2	2	2	-1	2	2	-2	2	0	0	0	0	-1	-1	-1	-2	-2	0	0	-1	-1	-1	1	1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₅	2	-2	2	-2	2	2	-2	0	0	0	0	0	0	2	-2	-2	2i	-2i	0	0	2	-2	0	0	0	0	2i	-2i	-2i	2i	complex lifted from C₄○D₄
ρ₁₆	2	-2	2	-2	2	2	-2	0	0	0	0	0	0	2	-2	-2	-2i	2i	0	0	2	-2	0	0	0	0	-2i	2i	2i	-2i	complex lifted from C₄○D₄
ρ₁₇	2	-2	2	-2	2	-2	2	0	0	0	0	0	0	2	-2	-2	0	0	-2i	2i	-2	2	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	2	-2	2	0	0	0	0	0	0	2	-2	-2	0	0	2i	-2i	-2	2	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	-1	2	-2	0	0	0	0	0	0	-1	1	1	2i	-2i	0	0	-1	1	-√3	-√-3	√-3	√3	-i	i	i	-i	complex lifted from C₄○D₁₂
ρ₂₀	2	-2	2	-2	-1	2	-2	0	0	0	0	0	0	-1	1	1	2i	-2i	0	0	-1	1	√3	√-3	-√-3	-√3	-i	i	i	-i	complex lifted from C₄○D₁₂
ρ₂₁	2	-2	2	-2	-1	2	-2	0	0	0	0	0	0	-1	1	1	-2i	2i	0	0	-1	1	√3	-√-3	√-3	-√3	i	-i	-i	i	complex lifted from C₄○D₁₂
ρ₂₂	2	-2	2	-2	-1	2	-2	0	0	0	0	0	0	-1	1	1	-2i	2i	0	0	-1	1	-√3	√-3	-√-3	√3	i	-i	-i	i	complex lifted from C₄○D₁₂
ρ₂₃	4	4	4	4	-2	-4	-4	0	0	0	0	0	0	-2	-2	-2	0	0	0	0	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from S₃×D₄
ρ₂₄	4	-4	4	-4	-2	-4	4	0	0	0	0	0	0	-2	2	2	0	0	0	0	2	-2	0	0	0	0	0	0	0	0	symplectic lifted from D₄⋊₂S₃, Schur index 2
ρ₂₅	4	-4	-4	4	4	0	0	0	0	0	0	0	0	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₆	4	4	-4	-4	4	0	0	0	0	0	0	0	0	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₇	4	4	-4	-4	-2	0	0	0	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	0	0	√6	-√6	√6	-√6	symplectic lifted from D₄.D₆, Schur index 2
ρ₂₈	4	4	-4	-4	-2	0	0	0	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	0	0	-√6	√6	-√6	√6	symplectic lifted from D₄.D₆, Schur index 2
ρ₂₉	4	-4	-4	4	-2	0	0	0	0	0	0	0	0	2	-2	2	0	0	0	0	0	0	0	0	0	0	√-6	√-6	-√-6	-√-6	complex lifted from Q₁₆⋊S₃
ρ₃₀	4	-4	-4	4	-2	0	0	0	0	0	0	0	0	2	-2	2	0	0	0	0	0	0	0	0	0	0	-√-6	-√-6	√-6	√-6	complex lifted from Q₁₆⋊S₃

Smallest permutation representation of (C₂×Q₈).₃₆D₆
►Regular action on 192 points

Generators in S₁₉₂

(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 49)(10 50)(11 51)(12 52)(13 122)(14 123)(15 124)(16 125)(17 126)(18 127)(19 128)(20 129)(21 130)(22 131)(23 132)(24 121)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 133)(34 134)(35 135)(36 136)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 157)(83 158)(84 159)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(91 191)(92 192)(93 181)(94 182)(95 183)(96 184)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 109)(106 110)(107 111)(108 112)(145 180)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)

(1 192 59 86)(2 87 60 181)(3 182 49 88)(4 89 50 183)(5 184 51 90)(6 91 52 185)(7 186 53 92)(8 93 54 187)(9 188 55 94)(10 95 56 189)(11 190 57 96)(12 85 58 191)(13 69 128 45)(14 46 129 70)(15 71 130 47)(16 48 131 72)(17 61 132 37)(18 38 121 62)(19 63 122 39)(20 40 123 64)(21 65 124 41)(22 42 125 66)(23 67 126 43)(24 44 127 68)(25 100 143 110)(26 111 144 101)(27 102 133 112)(28 113 134 103)(29 104 135 114)(30 115 136 105)(31 106 137 116)(32 117 138 107)(33 108 139 118)(34 119 140 97)(35 98 141 120)(36 109 142 99)(73 173 166 156)(74 145 167 174)(75 175 168 146)(76 147 157 176)(77 177 158 148)(78 149 159 178)(79 179 160 150)(80 151 161 180)(81 169 162 152)(82 153 163 170)(83 171 164 154)(84 155 165 172)

(1 84 59 165)(2 179 60 150)(3 74 49 167)(4 169 50 152)(5 76 51 157)(6 171 52 154)(7 78 53 159)(8 173 54 156)(9 80 55 161)(10 175 56 146)(11 82 57 163)(12 177 58 148)(13 36 128 142)(14 106 129 116)(15 26 130 144)(16 108 131 118)(17 28 132 134)(18 98 121 120)(19 30 122 136)(20 100 123 110)(21 32 124 138)(22 102 125 112)(23 34 126 140)(24 104 127 114)(25 64 143 40)(27 66 133 42)(29 68 135 44)(31 70 137 46)(33 72 139 48)(35 62 141 38)(37 113 61 103)(39 115 63 105)(41 117 65 107)(43 119 67 97)(45 109 69 99)(47 111 71 101)(73 187 166 93)(75 189 168 95)(77 191 158 85)(79 181 160 87)(81 183 162 89)(83 185 164 91)(86 155 192 172)(88 145 182 174)(90 147 184 176)(92 149 186 178)(94 151 188 180)(96 153 190 170)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)

(1 115 53 99)(2 104 54 120)(3 113 55 97)(4 102 56 118)(5 111 57 107)(6 100 58 116)(7 109 59 105)(8 98 60 114)(9 119 49 103)(10 108 50 112)(11 117 51 101)(12 106 52 110)(13 165 122 78)(14 83 123 158)(15 163 124 76)(16 81 125 168)(17 161 126 74)(18 79 127 166)(19 159 128 84)(20 77 129 164)(21 157 130 82)(22 75 131 162)(23 167 132 80)(24 73 121 160)(25 191 137 91)(26 96 138 184)(27 189 139 89)(28 94 140 182)(29 187 141 87)(30 92 142 192)(31 185 143 85)(32 90 144 190)(33 183 133 95)(34 88 134 188)(35 181 135 93)(36 86 136 186)(37 180 67 145)(38 150 68 173)(39 178 69 155)(40 148 70 171)(41 176 71 153)(42 146 72 169)(43 174 61 151)(44 156 62 179)(45 172 63 149)(46 154 64 177)(47 170 65 147)(48 152 66 175)

G:=sub<Sym(192)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,49)(10,50)(11,51)(12,52)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,132)(24,121)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,133)(34,134)(35,135)(36,136)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,157)(83,158)(84,159)(85,185)(86,186)(87,187)(88,188)(89,189)(90,190)(91,191)(92,192)(93,181)(94,182)(95,183)(96,184)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,109)(106,110)(107,111)(108,112)(145,180)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179), (1,192,59,86)(2,87,60,181)(3,182,49,88)(4,89,50,183)(5,184,51,90)(6,91,52,185)(7,186,53,92)(8,93,54,187)(9,188,55,94)(10,95,56,189)(11,190,57,96)(12,85,58,191)(13,69,128,45)(14,46,129,70)(15,71,130,47)(16,48,131,72)(17,61,132,37)(18,38,121,62)(19,63,122,39)(20,40,123,64)(21,65,124,41)(22,42,125,66)(23,67,126,43)(24,44,127,68)(25,100,143,110)(26,111,144,101)(27,102,133,112)(28,113,134,103)(29,104,135,114)(30,115,136,105)(31,106,137,116)(32,117,138,107)(33,108,139,118)(34,119,140,97)(35,98,141,120)(36,109,142,99)(73,173,166,156)(74,145,167,174)(75,175,168,146)(76,147,157,176)(77,177,158,148)(78,149,159,178)(79,179,160,150)(80,151,161,180)(81,169,162,152)(82,153,163,170)(83,171,164,154)(84,155,165,172), (1,84,59,165)(2,179,60,150)(3,74,49,167)(4,169,50,152)(5,76,51,157)(6,171,52,154)(7,78,53,159)(8,173,54,156)(9,80,55,161)(10,175,56,146)(11,82,57,163)(12,177,58,148)(13,36,128,142)(14,106,129,116)(15,26,130,144)(16,108,131,118)(17,28,132,134)(18,98,121,120)(19,30,122,136)(20,100,123,110)(21,32,124,138)(22,102,125,112)(23,34,126,140)(24,104,127,114)(25,64,143,40)(27,66,133,42)(29,68,135,44)(31,70,137,46)(33,72,139,48)(35,62,141,38)(37,113,61,103)(39,115,63,105)(41,117,65,107)(43,119,67,97)(45,109,69,99)(47,111,71,101)(73,187,166,93)(75,189,168,95)(77,191,158,85)(79,181,160,87)(81,183,162,89)(83,185,164,91)(86,155,192,172)(88,145,182,174)(90,147,184,176)(92,149,186,178)(94,151,188,180)(96,153,190,170), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,115,53,99)(2,104,54,120)(3,113,55,97)(4,102,56,118)(5,111,57,107)(6,100,58,116)(7,109,59,105)(8,98,60,114)(9,119,49,103)(10,108,50,112)(11,117,51,101)(12,106,52,110)(13,165,122,78)(14,83,123,158)(15,163,124,76)(16,81,125,168)(17,161,126,74)(18,79,127,166)(19,159,128,84)(20,77,129,164)(21,157,130,82)(22,75,131,162)(23,167,132,80)(24,73,121,160)(25,191,137,91)(26,96,138,184)(27,189,139,89)(28,94,140,182)(29,187,141,87)(30,92,142,192)(31,185,143,85)(32,90,144,190)(33,183,133,95)(34,88,134,188)(35,181,135,93)(36,86,136,186)(37,180,67,145)(38,150,68,173)(39,178,69,155)(40,148,70,171)(41,176,71,153)(42,146,72,169)(43,174,61,151)(44,156,62,179)(45,172,63,149)(46,154,64,177)(47,170,65,147)(48,152,66,175)>;

G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,49)(10,50)(11,51)(12,52)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,132)(24,121)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,133)(34,134)(35,135)(36,136)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,157)(83,158)(84,159)(85,185)(86,186)(87,187)(88,188)(89,189)(90,190)(91,191)(92,192)(93,181)(94,182)(95,183)(96,184)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,109)(106,110)(107,111)(108,112)(145,180)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179), (1,192,59,86)(2,87,60,181)(3,182,49,88)(4,89,50,183)(5,184,51,90)(6,91,52,185)(7,186,53,92)(8,93,54,187)(9,188,55,94)(10,95,56,189)(11,190,57,96)(12,85,58,191)(13,69,128,45)(14,46,129,70)(15,71,130,47)(16,48,131,72)(17,61,132,37)(18,38,121,62)(19,63,122,39)(20,40,123,64)(21,65,124,41)(22,42,125,66)(23,67,126,43)(24,44,127,68)(25,100,143,110)(26,111,144,101)(27,102,133,112)(28,113,134,103)(29,104,135,114)(30,115,136,105)(31,106,137,116)(32,117,138,107)(33,108,139,118)(34,119,140,97)(35,98,141,120)(36,109,142,99)(73,173,166,156)(74,145,167,174)(75,175,168,146)(76,147,157,176)(77,177,158,148)(78,149,159,178)(79,179,160,150)(80,151,161,180)(81,169,162,152)(82,153,163,170)(83,171,164,154)(84,155,165,172), (1,84,59,165)(2,179,60,150)(3,74,49,167)(4,169,50,152)(5,76,51,157)(6,171,52,154)(7,78,53,159)(8,173,54,156)(9,80,55,161)(10,175,56,146)(11,82,57,163)(12,177,58,148)(13,36,128,142)(14,106,129,116)(15,26,130,144)(16,108,131,118)(17,28,132,134)(18,98,121,120)(19,30,122,136)(20,100,123,110)(21,32,124,138)(22,102,125,112)(23,34,126,140)(24,104,127,114)(25,64,143,40)(27,66,133,42)(29,68,135,44)(31,70,137,46)(33,72,139,48)(35,62,141,38)(37,113,61,103)(39,115,63,105)(41,117,65,107)(43,119,67,97)(45,109,69,99)(47,111,71,101)(73,187,166,93)(75,189,168,95)(77,191,158,85)(79,181,160,87)(81,183,162,89)(83,185,164,91)(86,155,192,172)(88,145,182,174)(90,147,184,176)(92,149,186,178)(94,151,188,180)(96,153,190,170), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,115,53,99)(2,104,54,120)(3,113,55,97)(4,102,56,118)(5,111,57,107)(6,100,58,116)(7,109,59,105)(8,98,60,114)(9,119,49,103)(10,108,50,112)(11,117,51,101)(12,106,52,110)(13,165,122,78)(14,83,123,158)(15,163,124,76)(16,81,125,168)(17,161,126,74)(18,79,127,166)(19,159,128,84)(20,77,129,164)(21,157,130,82)(22,75,131,162)(23,167,132,80)(24,73,121,160)(25,191,137,91)(26,96,138,184)(27,189,139,89)(28,94,140,182)(29,187,141,87)(30,92,142,192)(31,185,143,85)(32,90,144,190)(33,183,133,95)(34,88,134,188)(35,181,135,93)(36,86,136,186)(37,180,67,145)(38,150,68,173)(39,178,69,155)(40,148,70,171)(41,176,71,153)(42,146,72,169)(43,174,61,151)(44,156,62,179)(45,172,63,149)(46,154,64,177)(47,170,65,147)(48,152,66,175) );

G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,49),(10,50),(11,51),(12,52),(13,122),(14,123),(15,124),(16,125),(17,126),(18,127),(19,128),(20,129),(21,130),(22,131),(23,132),(24,121),(25,137),(26,138),(27,139),(28,140),(29,141),(30,142),(31,143),(32,144),(33,133),(34,134),(35,135),(36,136),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(73,160),(74,161),(75,162),(76,163),(77,164),(78,165),(79,166),(80,167),(81,168),(82,157),(83,158),(84,159),(85,185),(86,186),(87,187),(88,188),(89,189),(90,190),(91,191),(92,192),(93,181),(94,182),(95,183),(96,184),(97,113),(98,114),(99,115),(100,116),(101,117),(102,118),(103,119),(104,120),(105,109),(106,110),(107,111),(108,112),(145,180),(146,169),(147,170),(148,171),(149,172),(150,173),(151,174),(152,175),(153,176),(154,177),(155,178),(156,179)], [(1,192,59,86),(2,87,60,181),(3,182,49,88),(4,89,50,183),(5,184,51,90),(6,91,52,185),(7,186,53,92),(8,93,54,187),(9,188,55,94),(10,95,56,189),(11,190,57,96),(12,85,58,191),(13,69,128,45),(14,46,129,70),(15,71,130,47),(16,48,131,72),(17,61,132,37),(18,38,121,62),(19,63,122,39),(20,40,123,64),(21,65,124,41),(22,42,125,66),(23,67,126,43),(24,44,127,68),(25,100,143,110),(26,111,144,101),(27,102,133,112),(28,113,134,103),(29,104,135,114),(30,115,136,105),(31,106,137,116),(32,117,138,107),(33,108,139,118),(34,119,140,97),(35,98,141,120),(36,109,142,99),(73,173,166,156),(74,145,167,174),(75,175,168,146),(76,147,157,176),(77,177,158,148),(78,149,159,178),(79,179,160,150),(80,151,161,180),(81,169,162,152),(82,153,163,170),(83,171,164,154),(84,155,165,172)], [(1,84,59,165),(2,179,60,150),(3,74,49,167),(4,169,50,152),(5,76,51,157),(6,171,52,154),(7,78,53,159),(8,173,54,156),(9,80,55,161),(10,175,56,146),(11,82,57,163),(12,177,58,148),(13,36,128,142),(14,106,129,116),(15,26,130,144),(16,108,131,118),(17,28,132,134),(18,98,121,120),(19,30,122,136),(20,100,123,110),(21,32,124,138),(22,102,125,112),(23,34,126,140),(24,104,127,114),(25,64,143,40),(27,66,133,42),(29,68,135,44),(31,70,137,46),(33,72,139,48),(35,62,141,38),(37,113,61,103),(39,115,63,105),(41,117,65,107),(43,119,67,97),(45,109,69,99),(47,111,71,101),(73,187,166,93),(75,189,168,95),(77,191,158,85),(79,181,160,87),(81,183,162,89),(83,185,164,91),(86,155,192,172),(88,145,182,174),(90,147,184,176),(92,149,186,178),(94,151,188,180),(96,153,190,170)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,115,53,99),(2,104,54,120),(3,113,55,97),(4,102,56,118),(5,111,57,107),(6,100,58,116),(7,109,59,105),(8,98,60,114),(9,119,49,103),(10,108,50,112),(11,117,51,101),(12,106,52,110),(13,165,122,78),(14,83,123,158),(15,163,124,76),(16,81,125,168),(17,161,126,74),(18,79,127,166),(19,159,128,84),(20,77,129,164),(21,157,130,82),(22,75,131,162),(23,167,132,80),(24,73,121,160),(25,191,137,91),(26,96,138,184),(27,189,139,89),(28,94,140,182),(29,187,141,87),(30,92,142,192),(31,185,143,85),(32,90,144,190),(33,183,133,95),(34,88,134,188),(35,181,135,93),(36,86,136,186),(37,180,67,145),(38,150,68,173),(39,178,69,155),(40,148,70,171),(41,176,71,153),(42,146,72,169),(43,174,61,151),(44,156,62,179),(45,172,63,149),(46,154,64,177),(47,170,65,147),(48,152,66,175)]])

Matrix representation of (C₂×Q₈).₃₆D₆ ►in GL₈(𝔽₇₃)

72	0	0	0	0	0	0	0
0	72	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	72	0
0	0	0	0	0	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	0	0	72
0	0	0	0	0	0	1	0

40	12	11	41	0	0	0	0
61	28	21	11	0	0	0	0
4	37	45	61	0	0	0	0
40	4	12	33	0	0	0	0
0	0	0	0	70	61	71	2
0	0	0	0	61	3	71	71
0	0	0	0	2	2	12	70
0	0	0	0	71	2	70	61

66	7	25	23	0	0	0	0
66	59	25	25	0	0	0	0
0	0	14	66	0	0	0	0
0	0	7	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0

30	43	62	69	0	0	0	0
13	43	69	15	0	0	0	0
0	0	20	55	0	0	0	0
0	0	2	53	0	0	0	0
0	0	0	0	24	8	35	0
0	0	0	0	8	49	0	35
0	0	0	0	38	0	49	65
0	0	0	0	0	38	65	24

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[40,61,4,40,0,0,0,0,12,28,37,4,0,0,0,0,11,21,45,12,0,0,0,0,41,11,61,33,0,0,0,0,0,0,0,0,70,61,2,71,0,0,0,0,61,3,2,2,0,0,0,0,71,71,12,70,0,0,0,0,2,71,70,61],[66,66,0,0,0,0,0,0,7,59,0,0,0,0,0,0,25,25,14,7,0,0,0,0,23,25,66,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[30,13,0,0,0,0,0,0,43,43,0,0,0,0,0,0,62,69,20,2,0,0,0,0,69,15,55,53,0,0,0,0,0,0,0,0,24,8,38,0,0,0,0,0,8,49,0,38,0,0,0,0,35,0,49,65,0,0,0,0,0,35,65,24] >;

(C₂×Q₈).₃₆D₆ in GAP, Magma, Sage, TeX

(C_2\times Q_8)._{36}D_6

% in TeX

G:=Group("(C2xQ8).36D6");

// GroupNames label

G:=SmallGroup(192,356);

// by ID

G=gap.SmallGroup(192,356);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,232,1094,135,100,570,297,136,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=b^2,d^6=a*b^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=a*b^-1*c,e*c*e^-1=b*c,e*d*e^-1=a*d^5>;

// generators/relations

Export

Character table of (C₂×Q₈).₃₆D₆ in TeX

72	0	0	0	0	0	0	0
0	72	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	72	0
0	0	0	0	0	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	0	0	72
0	0	0	0	0	0	1	0

40	12	11	41	0	0	0	0
61	28	21	11	0	0	0	0
4	37	45	61	0	0	0	0
40	4	12	33	0	0	0	0
0	0	0	0	70	61	71	2
0	0	0	0	61	3	71	71
0	0	0	0	2	2	12	70
0	0	0	0	71	2	70	61

66	7	25	23	0	0	0	0
66	59	25	25	0	0	0	0
0	0	14	66	0	0	0	0
0	0	7	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0

30	43	62	69	0	0	0	0
13	43	69	15	0	0	0	0
0	0	20	55	0	0	0	0
0	0	2	53	0	0	0	0
0	0	0	0	24	8	35	0
0	0	0	0	8	49	0	35
0	0	0	0	38	0	49	65
0	0	0	0	0	38	65	24

72	0	0	0	0	0	0	0
0	72	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	72	0
0	0	0	0	0	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	0	0	72
0	0	0	0	0	0	1	0

40	12	11	41	0	0	0	0
61	28	21	11	0	0	0	0
4	37	45	61	0	0	0	0
40	4	12	33	0	0	0	0
0	0	0	0	70	61	71	2
0	0	0	0	61	3	71	71
0	0	0	0	2	2	12	70
0	0	0	0	71	2	70	61

66	7	25	23	0	0	0	0
66	59	25	25	0	0	0	0
0	0	14	66	0	0	0	0
0	0	7	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0

30	43	62	69	0	0	0	0
13	43	69	15	0	0	0	0
0	0	20	55	0	0	0	0
0	0	2	53	0	0	0	0
0	0	0	0	24	8	35	0
0	0	0	0	8	49	0	35
0	0	0	0	38	0	49	65
0	0	0	0	0	38	65	24

G = (C2×Q8).36D6 order 192 = 26·3

12nd non-split extension by C2×Q8 of D6 acting via D6/C3=C22

G = (C₂×Q₈).₃₆D₆ order 192 = 2⁶·3

12^nd non-split extension by C₂×Q₈ of D₆ acting via D₆/C₃=C₂²

72	0	0	0	0	0	0	0
0	72	0	0	0	0	0	0
0	0	72	0	0	0	0	0
0	0	0	72	0	0	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	72	0
0	0	0	0	0	0	0	72

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	72	0	0	0
0	0	0	0	0	0	0	72
0	0	0	0	0	0	1	0

40	12	11	41	0	0	0	0
61	28	21	11	0	0	0	0
4	37	45	61	0	0	0	0
40	4	12	33	0	0	0	0
0	0	0	0	70	61	71	2
0	0	0	0	61	3	71	71
0	0	0	0	2	2	12	70
0	0	0	0	71	2	70	61

66	7	25	23	0	0	0	0
66	59	25	25	0	0	0	0
0	0	14	66	0	0	0	0
0	0	7	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0

30	43	62	69	0	0	0	0
13	43	69	15	0	0	0	0
0	0	20	55	0	0	0	0
0	0	2	53	0	0	0	0
0	0	0	0	24	8	35	0
0	0	0	0	8	49	0	35
0	0	0	0	38	0	49	65
0	0	0	0	0	38	65	24