(C2xD4).F5 - GroupNames

G = (C₂×D₄).F₅ order 320 = 2⁶·5

1^st non-split extension by C₂×D₄ of F₅ acting via F₅/C₅=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₀.₅C₄≀C₂, (C₂×D₄).₁F₅, (D₄×C₁₀).₁C₄, C₂.₇(D₄⋊F₅), (C₂×Dic₁₀).₅C₄, C₂.₄(C₂³.F₅), C₁₀.C₄²⋊₁C₂, (C₂×Dic₅).₁₀₆D₄, C₂₀.₁₇D₄.₂C₂, C₁₀.₃(C₄.D₄), (C₄×Dic₅).₃C₂², C₅⋊₁(C₄².C₂²), C₂².₆₀(C₂²⋊F₅), (C₂×C₄).₁₅(C₂×F₅), (C₂×C₂₀).₁₀(C₂×C₄), (C₂×C₁₀).₃₄(C₂²⋊C₄), SmallGroup(320,259)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — (C₂×D₄).F₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₄×Dic₅ — C₁₀.C₄² — (C₂×D₄).F₅

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀ — (C₂×D₄).F₅

Upper central

C₁ — C₂² — C₂×C₄ — C₂×D₄

Generators and relations for (C₂×D₄).F₅
G = < a,b,c,d,e | a²=b⁴=c²=d⁵=1, e⁴=a, ebe^-1=ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, cd=dc, ece^-1=b^-1c, ede^-1=d³ >

Subgroups: 298 in 70 conjugacy classes, 22 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₁₀, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂×C₈, C₂×D₄, C₂×Q₈, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₈⋊C₄, C₄.₄D₄, C₅⋊C₈, Dic₁₀, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₄².C₂², C₄×Dic₅, C₂³.D₅, C₂×C₅⋊C₈, C₂×Dic₁₀, D₄×C₁₀, C₁₀.C₄², C₂₀.₁₇D₄, (C₂×D₄).F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, F₅, C₄.D₄, C₄≀C₂, C₂×F₅, C₄².C₂², C₂²⋊F₅, D₄⋊F₅, C₂³.F₅, (C₂×D₄).F₅

Character table of (C₂×D₄).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	10D	10E	10F	10G	20A	20B
size	1	1	1	1	8	4	10	10	10	10	40	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	2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	2	2	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	-2	-2	-2	2	2	0	2	0	0	0	0	0	0	0	0	2	2	2	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	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
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	2i	-2i	0	0	0	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
ρ₁₄	2	2	-2	-2	0	0	-2i	2i	0	0	0	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
ρ₁₅	2	2	-2	-2	0	0	2i	-2i	0	0	0	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	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
ρ₁₈	2	2	-2	-2	0	0	-2i	2i	0	0	0	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	4	4	0	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	-4	4	0	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
ρ₂₁	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
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	-√5	1	1	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	√5	1	1	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+2ζ₅³+1	2ζ₅²+2ζ₅+1	2ζ₅⁴+2ζ₅²+1	2ζ₅³+2ζ₅+1	-√5	√5	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	2ζ₅²+2ζ₅+1	2ζ₅⁴+2ζ₅³+1	2ζ₅³+2ζ₅+1	2ζ₅⁴+2ζ₅²+1	-√5	√5	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	2ζ₅³+2ζ₅+1	2ζ₅⁴+2ζ₅²+1	2ζ₅⁴+2ζ₅³+1	2ζ₅²+2ζ₅+1	√5	-√5	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+2ζ₅²+1	2ζ₅³+2ζ₅+1	2ζ₅²+2ζ₅+1	2ζ₅⁴+2ζ₅³+1	√5	-√5	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	symplectic lifted from D₄⋊F₅, Schur index 2
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	symplectic lifted from D₄⋊F₅, Schur index 2

Smallest permutation representation of (C₂×D₄).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)

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

(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), (2,64)(3,141)(4,34)(6,60)(7,137)(8,38)(9,17)(10,87)(12,56)(13,21)(14,83)(16,52)(18,54)(20,81)(22,50)(24,85)(25,92)(26,153)(27,110)(29,96)(30,157)(31,106)(33,37)(35,63)(36,144)(39,59)(40,140)(41,65)(42,46)(43,76)(44,135)(45,69)(47,80)(48,131)(49,53)(51,88)(55,84)(57,61)(58,142)(62,138)(66,75)(67,130)(70,79)(71,134)(74,136)(78,132)(82,86)(89,93)(90,158)(91,111)(94,154)(95,107)(97,118)(98,126)(99,150)(101,114)(102,122)(103,146)(105,109)(108,160)(112,156)(113,147)(115,119)(116,123)(117,151)(120,127)(121,152)(125,148)(129,133)(145,149), (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), (2,64)(3,141)(4,34)(6,60)(7,137)(8,38)(9,17)(10,87)(12,56)(13,21)(14,83)(16,52)(18,54)(20,81)(22,50)(24,85)(25,92)(26,153)(27,110)(29,96)(30,157)(31,106)(33,37)(35,63)(36,144)(39,59)(40,140)(41,65)(42,46)(43,76)(44,135)(45,69)(47,80)(48,131)(49,53)(51,88)(55,84)(57,61)(58,142)(62,138)(66,75)(67,130)(70,79)(71,134)(74,136)(78,132)(82,86)(89,93)(90,158)(91,111)(94,154)(95,107)(97,118)(98,126)(99,150)(101,114)(102,122)(103,146)(105,109)(108,160)(112,156)(113,147)(115,119)(116,123)(117,151)(120,127)(121,152)(125,148)(129,133)(145,149), (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)], [(2,64),(3,141),(4,34),(6,60),(7,137),(8,38),(9,17),(10,87),(12,56),(13,21),(14,83),(16,52),(18,54),(20,81),(22,50),(24,85),(25,92),(26,153),(27,110),(29,96),(30,157),(31,106),(33,37),(35,63),(36,144),(39,59),(40,140),(41,65),(42,46),(43,76),(44,135),(45,69),(47,80),(48,131),(49,53),(51,88),(55,84),(57,61),(58,142),(62,138),(66,75),(67,130),(70,79),(71,134),(74,136),(78,132),(82,86),(89,93),(90,158),(91,111),(94,154),(95,107),(97,118),(98,126),(99,150),(101,114),(102,122),(103,146),(105,109),(108,160),(112,156),(113,147),(115,119),(116,123),(117,151),(120,127),(121,152),(125,148),(129,133),(145,149)], [(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×D₄).F₅ ►in GL₈(𝔽₄₁)

1	0	0	0	0	0	0	0
0	1	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	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	34	30	0	0	0	0
0	0	12	7	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

1	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	36	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

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

36	36	0	0	0	0	0	0
5	36	0	0	0	0	0	0
0	0	40	14	0	0	0	0
0	0	24	1	0	0	0	0
0	0	0	0	16	3	34	14
0	0	0	0	31	11	38	13
0	0	0	0	27	2	30	20
0	0	0	0	28	18	39	25

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,34,12,0,0,0,0,0,0,30,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[36,5,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,40,24,0,0,0,0,0,0,14,1,0,0,0,0,0,0,0,0,16,31,27,28,0,0,0,0,3,11,2,18,0,0,0,0,34,38,30,39,0,0,0,0,14,13,20,25] >;

(C₂×D₄).F₅ in GAP, Magma, Sage, TeX

(C_2\times D_4).F_5

% in TeX

G:=Group("(C2xD4).F5");

// GroupNames label

G:=SmallGroup(320,259);

// by ID

G=gap.SmallGroup(320,259);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=1,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=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×D₄).F₅ in TeX

1	0	0	0	0	0	0	0
0	1	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	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	34	30	0	0	0	0
0	0	12	7	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

1	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	36	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

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

36	36	0	0	0	0	0	0
5	36	0	0	0	0	0	0
0	0	40	14	0	0	0	0
0	0	24	1	0	0	0	0
0	0	0	0	16	3	34	14
0	0	0	0	31	11	38	13
0	0	0	0	27	2	30	20
0	0	0	0	28	18	39	25

1	0	0	0	0	0	0	0
0	1	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	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	34	30	0	0	0	0
0	0	12	7	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

1	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	36	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

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

36	36	0	0	0	0	0	0
5	36	0	0	0	0	0	0
0	0	40	14	0	0	0	0
0	0	24	1	0	0	0	0
0	0	0	0	16	3	34	14
0	0	0	0	31	11	38	13
0	0	0	0	27	2	30	20
0	0	0	0	28	18	39	25

G = (C2×D4).F5 order 320 = 26·5

1st non-split extension by C2×D4 of F5 acting via F5/C5=C4

G = (C₂×D₄).F₅ order 320 = 2⁶·5

1^st non-split extension by C₂×D₄ of F₅ acting via F₅/C₅=C₄

1	0	0	0	0	0	0	0
0	1	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	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	34	30	0	0	0	0
0	0	12	7	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

1	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	36	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

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

36	36	0	0	0	0	0	0
5	36	0	0	0	0	0	0
0	0	40	14	0	0	0	0
0	0	24	1	0	0	0	0
0	0	0	0	16	3	34	14
0	0	0	0	31	11	38	13
0	0	0	0	27	2	30	20
0	0	0	0	28	18	39	25