(C2^2xD5):Q8 - GroupNames

G = (C₂²×D₅)⋊Q₈ order 320 = 2⁶·5

2^nd semidirect product of C₂²×D₅ and Q₈ acting via Q₈/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂²×D₅)⋊₂Q₈, (C₂²×Q₈)⋊₂D₅, (C₂×C₂₀).₃₀₆D₄, C₅⋊₃(C₂³⋊Q₈), C₁₀.₇₇C₂²≀C₂, C₂².₅₃(Q₈×D₅), (C₂²×C₄).₁₆₀D₁₀, C₁₀.₈₁(C₂²⋊Q₈), C₂.₁₇(D₁₀⋊₃Q₈), C₁₀.₆₂(C₄.₄D₄), C₂.₁₁(C₂⁴⋊₂D₅), (C₂³×D₅).₂₆C₂², C₂³.₃₈₇(C₂²×D₅), C₁₀.₁₀C₄²⋊₄₇C₂, C₂.₁₄(C₂₀.₂₃D₄), (C₂²×C₁₀).₃₇₁C₂³, (C₂²×C₂₀).₃₉₉C₂², C₂².₅₆(Q₈⋊₂D₅), (C₂²×Dic₅).₇₁C₂², (Q₈×C₂×C₁₀)⋊₁₂C₂, (C₂×C₁₀).₈₉(C₂×Q₈), (C₂×C₁₀).₅₆₄(C₂×D₄), (C₂×C₄).₈₉(C₅⋊D₄), C₂².₂₂₃(C₂×C₅⋊D₄), (C₂×D₁₀⋊C₄).₂₇C₂, (C₂×C₁₀).₁₉₅(C₄○D₄), SmallGroup(320,858)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — (C₂²×D₅)⋊Q₈

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂³×D₅ — C₂×D₁₀⋊C₄ — (C₂²×D₅)⋊Q₈

Lower central

C₅ — C₂²×C₁₀ — (C₂²×D₅)⋊Q₈

Upper central

C₁ — C₂³ — C₂²×Q₈

Generators and relations for (C₂²×D₅)⋊Q₈
G = < a,b,c,d,e,f | a²=b²=c⁵=d²=e⁴=1, f²=e², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, ede^-1=abd, fef^-1=e^-1 >

Subgroups: 774 in 202 conjugacy classes, 65 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂×C₂²⋊C₄, C₂²×Q₈, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂³⋊Q₈, D₁₀⋊C₄, C₂²×Dic₅, C₂²×C₂₀, Q₈×C₁₀, C₂³×D₅, C₁₀.₁₀C₄², C₂×D₁₀⋊C₄, Q₈×C₂×C₁₀, (C₂²×D₅)⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₅, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₀, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₅⋊D₄, C₂²×D₅, C₂³⋊Q₈, Q₈×D₅, Q₈⋊₂D₅, C₂×C₅⋊D₄, D₁₀⋊₃Q₈, C₂₀.₂₃D₄, C₂⁴⋊₂D₅, (C₂²×D₅)⋊Q₈

Smallest permutation representation of (C₂²×D₅)⋊Q₈
►On 160 points

Generators in S₁₆₀

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

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

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

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

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

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

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

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

62 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4F	4G	···	4L	5A	5B	10A	···	10N	20A	···	20X
order	1	2	···	2	2	2	4	···	4	4	···	4	5	5	10	···	10	20	···	20
size	1	1	···	1	20	20	4	···	4	20	···	20	2	2	2	···	2	4	···	4

62 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	-	+		+		-	+
image	C₁	C₂	C₂	C₂	D₄	Q₈	D₅	C₄○D₄	D₁₀	C₅⋊D₄	Q₈×D₅	Q₈⋊₂D₅
kernel	(C₂²×D₅)⋊Q₈	C₁₀.₁₀C₄²	C₂×D₁₀⋊C₄	Q₈×C₂×C₁₀	C₂×C₂₀	C₂²×D₅	C₂²×Q₈	C₂×C₁₀	C₂²×C₄	C₂×C₄	C₂²	C₂²
# reps	1	3	3	1	6	2	2	6	6	24	2	6

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

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	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

35	40	0	0	0	0
36	40	0	0	0	0
0	0	40	40	0	0
0	0	36	35	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	34	0	0	0	0
35	0	0	0	0	0
0	0	0	7	0	0
0	0	6	0	0	0
0	0	0	0	40	0
0	0	0	0	9	1

23	1	0	0	0	0
5	18	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	9	2
0	0	0	0	0	32

40	0	0	0	0	0
0	40	0	0	0	0
0	0	23	40	0	0
0	0	36	18	0	0
0	0	0	0	9	0
0	0	0	0	1	32

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,36,0,0,0,0,40,40,0,0,0,0,0,0,40,36,0,0,0,0,40,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,35,0,0,0,0,34,0,0,0,0,0,0,0,0,6,0,0,0,0,7,0,0,0,0,0,0,0,40,9,0,0,0,0,0,1],[23,5,0,0,0,0,1,18,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,2,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,23,36,0,0,0,0,40,18,0,0,0,0,0,0,9,1,0,0,0,0,0,32] >;

(C₂²×D₅)⋊Q₈ in GAP, Magma, Sage, TeX

(C_2^2\times D_5)\rtimes Q_8

% in TeX

G:=Group("(C2^2xD5):Q8");

// GroupNames label

G:=SmallGroup(320,858);

// by ID

G=gap.SmallGroup(320,858);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,254,387,184,12550]);

// Polycyclic

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

// generators/relations

G = (C22×D5)⋊Q8 order 320 = 26·5

2nd semidirect product of C22×D5 and Q8 acting via Q8/C2=C22

G = (C₂²×D₅)⋊Q₈ order 320 = 2⁶·5

2^nd semidirect product of C₂²×D₅ and Q₈ acting via Q₈/C₂=C₂²