(C2xQ8).F5 - GroupNames

G = (C₂×Q₈).F₅ order 320 = 2⁶·5

1^st non-split extension by C₂×Q₈ of F₅ acting via F₅/C₅=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₀.₁₁C₄≀C₂, (C₂×Q₈).₁F₅, (C₂×D₂₀).₆C₄, (Q₈×C₁₀).₁C₄, C₂.₇(Q₈⋊₂F₅), C₂.₆(C₂³.F₅), C₁₀.C₄²⋊₂C₂, (C₂×Dic₅).₁₁₀D₄, C₂₀.₂₃D₄.₂C₂, C₁₀.₅(C₄.D₄), (C₄×Dic₅).₅C₂², C₅⋊₂(C₄².C₂²), C₂².₆₃(C₂²⋊F₅), (C₂×C₄).₁₈(C₂×F₅), (C₂×C₂₀).₁₅(C₂×C₄), (C₂×C₁₀).₄₀(C₂²⋊C₄), SmallGroup(320,265)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — (C₂×Q₈).F₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₄×Dic₅ — C₁₀.C₄² — (C₂×Q₈).F₅

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀ — (C₂×Q₈).F₅

Upper central

C₁ — C₂² — C₂×C₄ — C₂×Q₈

Generators and relations for (C₂×Q₈).F₅
G = < a,b,c,d,e | a²=b⁴=d⁵=1, c²=b², e⁴=a, ebe^-1=ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, cd=dc, ece^-1=b^-1c, ede^-1=d³ >

Subgroups: 362 in 70 conjugacy classes, 22 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂×C₈, C₂×D₄, C₂×Q₈, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₈⋊C₄, C₄.₄D₄, C₅⋊C₈, D₂₀, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₂²×D₅, C₄².C₂², C₄×Dic₅, D₁₀⋊C₄, C₂×C₅⋊C₈, C₂×D₂₀, Q₈×C₁₀, C₁₀.C₄², C₂₀.₂₃D₄, (C₂×Q₈).F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, F₅, C₄.D₄, C₄≀C₂, C₂×F₅, C₄².C₂², C₂²⋊F₅, Q₈⋊₂F₅, C₂³.F₅, (C₂×Q₈).F₅

Character table of (C₂×Q₈).F₅

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	5	8A	8B	8C	8D	8E	8F	8G	8H	10A	10B	10C	20A	20B	20C	20D	20E	20F
size	1	1	1	1	40	4	8	10	10	10	10	4	20	20	20	20	20	20	20	20	4	4	4	8	8	8	8	8	8
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	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	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	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	linear of order 2
ρ₅	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	i	i	i	-i	-i	-i	-i	i	1	1	1	-1	-1	-1	1	1	-1	linear of order 4
ρ₆	1	1	1	1	-1	1	1	-1	-1	-1	-1	1	i	-i	-i	i	i	-i	-i	i	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₇	1	1	1	1	-1	1	1	-1	-1	-1	-1	1	-i	i	i	-i	-i	i	i	-i	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₈	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	-i	-i	-i	i	i	i	i	-i	1	1	1	-1	-1	-1	1	1	-1	linear of order 4
ρ₉	2	2	2	2	0	-2	0	2	-2	-2	2	2	0	0	0	0	0	0	0	0	2	2	2	0	0	0	-2	-2	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	-2	0	-2	2	2	-2	2	0	0	0	0	0	0	0	0	2	2	2	0	0	0	-2	-2	0	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	0	0	0	2i	0	0	-2i	2	0	1+i	-1-i	1-i	-1+i	0	0	0	-2	2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₂	2	-2	2	-2	0	0	0	0	-2i	2i	0	2	-1+i	0	0	0	0	1+i	-1-i	1-i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₃	2	-2	2	-2	0	0	0	0	2i	-2i	0	2	1+i	0	0	0	0	-1+i	1-i	-1-i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₄	2	-2	2	-2	0	0	0	0	2i	-2i	0	2	-1-i	0	0	0	0	1-i	-1+i	1+i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₅	2	-2	-2	2	0	0	0	-2i	0	0	2i	2	0	-1+i	1-i	-1-i	1+i	0	0	0	-2	2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₆	2	-2	2	-2	0	0	0	0	-2i	2i	0	2	1-i	0	0	0	0	-1-i	1+i	-1+i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₇	2	-2	-2	2	0	0	0	2i	0	0	-2i	2	0	-1-i	1+i	-1+i	1-i	0	0	0	-2	2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₈	2	-2	-2	2	0	0	0	-2i	0	0	2i	2	0	1-i	-1+i	1+i	-1-i	0	0	0	-2	2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₉	4	4	4	4	0	4	4	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₂₀	4	4	-4	-4	0	0	0	0	0	0	0	4	0	0	0	0	0	0	0	0	-4	-4	4	0	0	0	0	0	0	orthogonal lifted from C₄.D₄
ρ₂₁	4	4	4	4	0	4	-4	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	1	1	1	-1	-1	1	orthogonal lifted from C₂×F₅
ρ₂₂	4	4	4	4	0	-4	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	-√5	-√5	√5	1	1	√5	orthogonal lifted from C₂²⋊F₅
ρ₂₃	4	4	4	4	0	-4	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	√5	√5	-√5	1	1	-√5	orthogonal lifted from C₂²⋊F₅
ρ₂₄	4	4	-4	-4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	1	-1	2ζ₅⁴+2ζ₅²+1	2ζ₅³+2ζ₅+1	2ζ₅⁴+2ζ₅³+1	-√5	√5	2ζ₅²+2ζ₅+1	complex lifted from C₂³.F₅
ρ₂₅	4	4	-4	-4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	1	-1	2ζ₅³+2ζ₅+1	2ζ₅⁴+2ζ₅²+1	2ζ₅²+2ζ₅+1	-√5	√5	2ζ₅⁴+2ζ₅³+1	complex lifted from C₂³.F₅
ρ₂₆	4	4	-4	-4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	1	-1	2ζ₅²+2ζ₅+1	2ζ₅⁴+2ζ₅³+1	2ζ₅⁴+2ζ₅²+1	√5	-√5	2ζ₅³+2ζ₅+1	complex lifted from C₂³.F₅
ρ₂₇	4	4	-4	-4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	1	-1	2ζ₅⁴+2ζ₅³+1	2ζ₅²+2ζ₅+1	2ζ₅³+2ζ₅+1	√5	-√5	2ζ₅⁴+2ζ₅²+1	complex lifted from C₂³.F₅
ρ₂₈	8	-8	8	-8	0	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	orthogonal lifted from Q₈⋊₂F₅
ρ₂₉	8	-8	-8	8	0	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	0	2	-2	2	0	0	0	0	0	0	orthogonal lifted from Q₈⋊₂F₅

Smallest permutation representation of (C₂×Q₈).F₅
►On 160 points

Generators in S₁₆₀

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

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

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

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

(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)

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

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

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

Matrix representation of (C₂×Q₈).F₅ ►in GL₈(𝔽₄₁)

40	0	0	0	0	0	0	0
0	40	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	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

0	1	0	0	0	0	0	0
40	0	0	0	0	0	0	0
0	0	0	40	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	40	0	0
0	0	0	0	0	0	40	0
0	0	0	0	0	0	0	40

32	0	0	0	0	0	0	0
0	9	0	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	32	0	0	0	0	0
0	0	0	0	19	3	0	38
0	0	0	0	0	22	3	38
0	0	0	0	38	3	22	0
0	0	0	0	38	0	3	19

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	0	0	40
0	0	0	0	1	0	0	40
0	0	0	0	0	1	0	40
0	0	0	0	0	0	1	40

36	5	0	0	0	0	0	0
5	5	0	0	0	0	0	0
0	0	37	37	0	0	0	0
0	0	4	37	0	0	0	0
0	0	0	0	34	34	4	6
0	0	0	0	38	40	8	40
0	0	0	0	1	33	1	3
0	0	0	0	35	37	7	7

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,19,0,38,38,0,0,0,0,3,22,3,0,0,0,0,0,0,3,22,3,0,0,0,0,38,38,0,19],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[36,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,37,4,0,0,0,0,0,0,37,37,0,0,0,0,0,0,0,0,34,38,1,35,0,0,0,0,34,40,33,37,0,0,0,0,4,8,1,7,0,0,0,0,6,40,3,7] >;

(C₂×Q₈).F₅ in GAP, Magma, Sage, TeX

(C_2\times Q_8).F_5

% in TeX

G:=Group("(C2xQ8).F5");

// GroupNames label

G:=SmallGroup(320,265);

// by ID

G=gap.SmallGroup(320,265);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,268,1571,570,136,6278,3156]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^5=1,c^2=b^2,e^4=a,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;

// generators/relations

Export

Character table of (C₂×Q₈).F₅ in TeX

40	0	0	0	0	0	0	0
0	40	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	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

0	1	0	0	0	0	0	0
40	0	0	0	0	0	0	0
0	0	0	40	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	40	0	0
0	0	0	0	0	0	40	0
0	0	0	0	0	0	0	40

32	0	0	0	0	0	0	0
0	9	0	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	32	0	0	0	0	0
0	0	0	0	19	3	0	38
0	0	0	0	0	22	3	38
0	0	0	0	38	3	22	0
0	0	0	0	38	0	3	19

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	0	0	40
0	0	0	0	1	0	0	40
0	0	0	0	0	1	0	40
0	0	0	0	0	0	1	40

36	5	0	0	0	0	0	0
5	5	0	0	0	0	0	0
0	0	37	37	0	0	0	0
0	0	4	37	0	0	0	0
0	0	0	0	34	34	4	6
0	0	0	0	38	40	8	40
0	0	0	0	1	33	1	3
0	0	0	0	35	37	7	7

40	0	0	0	0	0	0	0
0	40	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	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

0	1	0	0	0	0	0	0
40	0	0	0	0	0	0	0
0	0	0	40	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	40	0	0
0	0	0	0	0	0	40	0
0	0	0	0	0	0	0	40

32	0	0	0	0	0	0	0
0	9	0	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	32	0	0	0	0	0
0	0	0	0	19	3	0	38
0	0	0	0	0	22	3	38
0	0	0	0	38	3	22	0
0	0	0	0	38	0	3	19

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	0	0	40
0	0	0	0	1	0	0	40
0	0	0	0	0	1	0	40
0	0	0	0	0	0	1	40

36	5	0	0	0	0	0	0
5	5	0	0	0	0	0	0
0	0	37	37	0	0	0	0
0	0	4	37	0	0	0	0
0	0	0	0	34	34	4	6
0	0	0	0	38	40	8	40
0	0	0	0	1	33	1	3
0	0	0	0	35	37	7	7

G = (C2×Q8).F5 order 320 = 26·5

1st non-split extension by C2×Q8 of F5 acting via F5/C5=C4

G = (C₂×Q₈).F₅ order 320 = 2⁶·5

1^st non-split extension by C₂×Q₈ of F₅ acting via F₅/C₅=C₄

40	0	0	0	0	0	0	0
0	40	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	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

0	1	0	0	0	0	0	0
40	0	0	0	0	0	0	0
0	0	0	40	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	40	0	0
0	0	0	0	0	0	40	0
0	0	0	0	0	0	0	40

32	0	0	0	0	0	0	0
0	9	0	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	32	0	0	0	0	0
0	0	0	0	19	3	0	38
0	0	0	0	0	22	3	38
0	0	0	0	38	3	22	0
0	0	0	0	38	0	3	19

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	0	0	40
0	0	0	0	1	0	0	40
0	0	0	0	0	1	0	40
0	0	0	0	0	0	1	40

36	5	0	0	0	0	0	0
5	5	0	0	0	0	0	0
0	0	37	37	0	0	0	0
0	0	4	37	0	0	0	0
0	0	0	0	34	34	4	6
0	0	0	0	38	40	8	40
0	0	0	0	1	33	1	3
0	0	0	0	35	37	7	7