(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^[×2], C₂, C₄^[×4], C₂², C₂²^[×3], C₅, C₈^[×4], C₂×C₄, C₂×C₄^[×3], D₄, Q₈, C₂³, C₁₀, C₁₀^[×2], C₁₀, C₄², C₂²⋊C₄^[×2], C₂×C₈^[×2], C₂×D₄, C₂×Q₈, Dic₅^[×3], C₂₀, C₂×C₁₀, C₂×C₁₀^[×3], C₈⋊C₄^[×2], C₄.₄D₄, C₅⋊C₈^[×4], Dic₁₀, C₂×Dic₅^[×2], C₂×Dic₅, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₄².C₂², C₄×Dic₅, C₂³.D₅^[×2], C₂×C₅⋊C₈^[×2], C₂×Dic₁₀, D₄×C₁₀, C₁₀.C₄²^[×2], C₂₀.₁₇D₄, (C₂×D₄).F₅
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², C₂×C₄, D₄^[×2], C₂²⋊C₄, F₅, C₄.D₄, C₄≀C₂^[×2], C₂×F₅, C₄².C₂², C₂²⋊F₅, D₄⋊F₅^[×2], 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)

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

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

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

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