C2^2:Q8:25D5 - GroupNames

G = C₂²⋊Q₈⋊₂₅D₅ order 320 = 2⁶·5

1^st semidirect product of C₂²⋊Q₈ and D₅ acting through Inn(C₂²⋊Q₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂²⋊Q₈⋊₂₅D₅, C₄⋊C₄.₁₈₇D₁₀, (Q₈×Dic₅)⋊₁₁C₂, D₂₀⋊₈C₄⋊₂₃C₂, C₂₀⋊₇D₄.₁₅C₂, (C₂×Q₈).₁₂₂D₁₀, C₂²⋊C₄.₅₄D₁₀, C₄.Dic₁₀⋊₂₂C₂, Dic₅⋊₄D₄⋊₁₃C₂, C₂₀.₂₀₈(C₄○D₄), C₄.₇₁(D₄⋊₂D₅), C₂₀.₂₃D₄⋊₁₁C₂, (C₂×C₂₀).₅₀₂C₂³, (C₂×C₁₀).₁₆₈C₂⁴, (C₂²×C₄).₃₇₀D₁₀, Dic₅.₇₃(C₄○D₄), (C₂×D₂₀).₁₅₃C₂², C₂².D₂₀⋊₁₅C₂, C₄⋊Dic₅.₂₁₂C₂², (Q₈×C₁₀).₁₀₃C₂², C₂².₃(Q₈⋊₂D₅), (C₂²×D₅).₇₃C₂³, C₂³.₁₈₆(C₂²×D₅), C₂².₁₈₉(C₂³×D₅), D₁₀⋊C₄.₁₉C₂², (C₂²×C₂₀).₂₄₈C₂², (C₂²×C₁₀).₁₉₆C₂³, C₅⋊₇(C₂³.₃₆C₂³), (C₄×Dic₅).₁₁₀C₂², (C₂×Dic₅).₂₄₂C₂³, C₁₀.D₄.₂₃C₂², (C₂²×Dic₅).₂₄₆C₂², (C₂×C₄×Dic₅)⋊₉C₂, C₂.₄₆(D₅×C₄○D₄), C₄⋊C₄⋊₇D₅⋊₂₃C₂, C₄⋊C₄⋊D₅⋊₁₆C₂, (C₅×C₂²⋊Q₈)⋊₅C₂, C₁₀.₁₅₈(C₂×C₄○D₄), C₂.₄₄(C₂×D₄⋊₂D₅), C₂.₁₅(C₂×Q₈⋊₂D₅), (C₂×C₄×D₅).₁₀₀C₂², (C₂×C₄).₄₄(C₂²×D₅), (C₂×C₁₀).₂₅(C₄○D₄), (C₅×C₄⋊C₄).₁₅₄C₂², (C₂×C₅⋊D₄).₃₇C₂², (C₅×C₂²⋊C₄).₂₃C₂², SmallGroup(320,1296)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂²⋊Q₈⋊₂₅D₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₂²×Dic₅ — C₂×C₄×Dic₅ — C₂²⋊Q₈⋊₂₅D₅

Lower central

C₅ — C₂×C₁₀ — C₂²⋊Q₈⋊₂₅D₅

Upper central

C₁ — C₂² — C₂²⋊Q₈

Generators and relations for C₂²⋊Q₈⋊₂₅D₅
G = < a,b,c,d,e,f | a²=b²=c⁴=e⁵=f²=1, d²=c², dad^-1=faf=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, bf=fb, dcd^-1=fcf=c^-1, ce=ec, de=ed, fdf=bc²d, fef=e^-1 >

Subgroups: 766 in 234 conjugacy classes, 101 normal (43 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₄², C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄×D₅, D₂₀, C₂×Dic₅, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₂²×D₅, C₂²×C₁₀, C₂³.₃₆C₂³, C₄×Dic₅, C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, C₄⋊Dic₅, D₁₀⋊C₄, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₅×C₄⋊C₄, C₂×C₄×D₅, C₂×D₂₀, C₂²×Dic₅, C₂×C₅⋊D₄, C₂²×C₂₀, Q₈×C₁₀, Dic₅⋊₄D₄, C₂².D₂₀, C₄.Dic₁₀, C₄⋊C₄⋊₇D₅, D₂₀⋊₈C₄, C₄⋊C₄⋊D₅, C₂×C₄×Dic₅, C₂₀⋊₇D₄, Q₈×Dic₅, C₂₀.₂₃D₄, C₅×C₂²⋊Q₈, C₂²⋊Q₈⋊₂₅D₅
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄○D₄, C₂⁴, D₁₀, C₂×C₄○D₄, C₂²×D₅, C₂³.₃₆C₂³, D₄⋊₂D₅, Q₈⋊₂D₅, C₂³×D₅, C₂×D₄⋊₂D₅, C₂×Q₈⋊₂D₅, D₅×C₄○D₄, C₂²⋊Q₈⋊₂₅D₅

Smallest permutation representation of C₂²⋊Q₈⋊₂₅D₅
►On 160 points

Generators in S₁₆₀

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

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

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

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

(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)

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

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

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

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

56 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	···	4R	4S	4T	5A	5B	10A	···	10F	10G	10H	10I	10J	20A	···	20H	20I	···	20P
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	4	···	4	4	4	5	5	10	···	10	10	10	10	10	20	···	20	20	···	20
size	1	1	1	1	2	2	20	20	2	2	2	2	4	4	4	4	5	5	5	5	10	···	10	20	20	2	2	2	···	2	4	4	4	4	4	···	4	8	···	8

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+				+	+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₅	C₄○D₄	C₄○D₄	C₄○D₄	D₁₀	D₁₀	D₁₀	D₁₀	D₄⋊₂D₅	Q₈⋊₂D₅	D₅×C₄○D₄
kernel	C₂²⋊Q₈⋊₂₅D₅	Dic₅⋊₄D₄	C₂².D₂₀	C₄.Dic₁₀	C₄⋊C₄⋊₇D₅	D₂₀⋊₈C₄	C₄⋊C₄⋊D₅	C₂×C₄×Dic₅	C₂₀⋊₇D₄	Q₈×Dic₅	C₂₀.₂₃D₄	C₅×C₂²⋊Q₈	C₂²⋊Q₈	Dic₅	C₂₀	C₂×C₁₀	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₄	C₂²	C₂
# reps	1	2	2	1	2	1	2	1	1	1	1	1	2	4	4	4	4	6	2	2	4	4	4

Matrix representation of C₂²⋊Q₈⋊₂₅D₅ ►in GL₆(𝔽₄₁)

0	1	0	0	0	0
1	0	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	32	39
0	0	0	0	40	9

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

0	32	0	0	0	0
32	0	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	0	0	0	0	0
0	32	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	18
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	6	1	0	0
0	0	40	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	40	0	0	0	0
0	0	1	6	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	32	40

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,40,0,0,0,0,39,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,18,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,0,0,0,1,32,0,0,0,0,0,40] >;

C₂²⋊Q₈⋊₂₅D₅ in GAP, Magma, Sage, TeX

C_2^2\rtimes Q_8\rtimes_{25}D_5

% in TeX

G:=Group("C2^2:Q8:25D5");

// GroupNames label

G:=SmallGroup(320,1296);

// by ID

G=gap.SmallGroup(320,1296);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,794,297,136,12550]);

// Polycyclic

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

// generators/relations

G = C22⋊Q8⋊25D5 order 320 = 26·5

1st semidirect product of C22⋊Q8 and D5 acting through Inn(C22⋊Q8)

G = C₂²⋊Q₈⋊₂₅D₅ order 320 = 2⁶·5

1^st semidirect product of C₂²⋊Q₈ and D₅ acting through Inn(C₂²⋊Q₈)