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G = C22⋊Q8⋊D5  order 320 = 26·5

4th semidirect product of C22⋊Q8 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5⋊2C8⋊6D4, C5⋊5(C8⋊D4), C22⋊Q8⋊4D5, C4⋊C4.68D10, C4.175(D4×D5), (C2×C20).78D4, C20.155(C2×D4), (C2×Q8).30D10, D20⋊6C4⋊40C2, C20⋊7D4.13C2, Q8⋊Dic5⋊16C2, C10.D8⋊40C2, (C22×C10).95D4, C20.191(C4○D4), C4.64(D4⋊2D5), C10.98(C4⋊D4), (C2×C20).368C23, (C22×C4).129D10, C23.28(C5⋊D4), (Q8×C10).48C22, C2.16(D4⋊D10), C10.117(C8⋊C22), (C2×D20).104C22, C10.91(C8.C22), C4⋊Dic5.147C22, C2.19(Dic5⋊D4), C2.12(C20.C23), (C22×C20).172C22, (C2×Q8⋊D5)⋊11C2, (C5×C22⋊Q8)⋊4C2, (C2×C10).499(C2×D4), (C2×C4).56(C5⋊D4), (C2×C4.Dic5)⋊13C2, (C5×C4⋊C4).115C22, (C2×C4).468(C22×D5), C22.174(C2×C5⋊D4), (C2×C5⋊2C8).116C22, SmallGroup(320,676)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C20 — C22⋊Q8⋊D5
Chief series
C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊7D4 — C22⋊Q8⋊D5
Lower central
C5 — C10 — C2×C20 — C22⋊Q8⋊D5
Upper central
C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for C22⋊Q8⋊D5
 G = < a,b,c,d,e,f | a2=b2=c4=e5=f2=1, d2=c2, dad-1=ab=ba, ac=ca, ae=ea, faf=abc2, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, fdf=c-1d, fef=e-1 >

Subgroups: 502 in 120 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C5⋊2C8, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, C8⋊D4, C2×C5⋊2C8, C4.Dic5, C4⋊Dic5, D10⋊C4, Q8⋊D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, Q8×C10, C10.D8, D20⋊6C4, Q8⋊Dic5, C2×C4.Dic5, C20⋊7D4, C2×Q8⋊D5, C5×C22⋊Q8, C22⋊Q8⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C8⋊D4, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊D4, C20.C23, D4⋊D10, C22⋊Q8⋊D5

Smallest permutation representation of C22⋊Q8⋊D5
►On 160 points
Generators in S160
(1 89)(2 90)(3 86)(4 87)(5 88)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)

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

(1 49 9 44)(2 50 10 45)(3 46 6 41)(4 47 7 42)(5 48 8 43)(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)(81 131 86 136)(82 132 87 137)(83 133 88 138)(84 134 89 139)(85 135 90 140)(91 121 96 126)(92 122 97 127)(93 123 98 128)(94 124 99 129)(95 125 100 130)(101 151 106 156)(102 152 107 157)(103 153 108 158)(104 154 109 159)(105 155 110 160)(111 141 116 146)(112 142 117 147)(113 143 118 148)(114 144 119 149)(115 145 120 150)

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20P
order12222244444455888810···101010101020···2020···20
size1111440224884022202020202···244444···48···8

44 irreducible representations

dim111111112222222222444444
type++++++++++++++++-+-+
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4C5⋊D4C8⋊C22C8.C22D4×D5D4⋊2D5C20.C23D4⋊D10
kernelC22⋊Q8⋊D5C10.D8D20⋊6C4Q8⋊Dic5C2×C4.Dic5C20⋊7D4C2×Q8⋊D5C5×C22⋊Q8C5⋊2C8C2×C20C22×C10C22⋊Q8C20C4⋊C4C22×C4C2×Q8C2×C4C23C10C10C4C4C2C2
# reps111111112112222244112244

Matrix representation of C22⋊Q8⋊D5 ►in GL8(𝔽41)

401430270000
27130300000
381540270000
3381410000
0000174000
000012400
0000002340
0000003618
,
400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
400000000
040000000
004000000
000400000
000040033
0000040380
000002810
0000131301
,
23611140000
351811110000
0023350000
006180000
0000288175
00003313512
00003082033
0000822121
,
01000000
406000000
00610000
004000000
00000100
0000403400
000000351
000000540
,
01000000
10000000
0040350000
00010000
00000100
00001000
000013011
00002828040

G:=sub<GL(8,GF(41))| [40,27,38,3,0,0,0,0,14,1,15,38,0,0,0,0,30,30,40,14,0,0,0,0,27,30,27,1,0,0,0,0,0,0,0,0,17,1,0,0,0,0,0,0,40,24,0,0,0,0,0,0,0,0,23,36,0,0,0,0,0,0,40,18],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,13,0,0,0,0,0,40,28,13,0,0,0,0,3,38,1,0,0,0,0,0,3,0,0,1],[23,35,0,0,0,0,0,0,6,18,0,0,0,0,0,0,11,11,23,6,0,0,0,0,14,11,35,18,0,0,0,0,0,0,0,0,28,33,30,8,0,0,0,0,8,13,8,22,0,0,0,0,17,5,20,1,0,0,0,0,5,12,33,21],[0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,35,5,0,0,0,0,0,0,1,40],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,1,0,0,0,0,0,0,0,0,0,1,13,28,0,0,0,0,1,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40] >;
 

C22⋊Q8⋊D5 in GAP, Magma, Sage, TeX 

C_2^2\rtimes Q_8\rtimes D_5
 
% in TeX
 
G:=Group("C2^2:Q8:D5");
 
// GroupNames label
 
G:=SmallGroup(320,676);
 
// by ID
 
G=gap.SmallGroup(320,676);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,555,184,1123,297,136,12550]);
 
// 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=a*b=b*a,a*c=c*a,a*e=e*a,f*a*f=a*b*c^2,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=c^-1*d,f*e*f=e^-1>;
 
// generators/relations
 

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