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G = C22×C40⋊C2order 320 = 26·5

Direct product of C22 and C40⋊C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×C40⋊C2, C4010C23, C20.54C24, C23.60D20, Dic104C23, D20.20C23, (C2×C8)⋊35D10, C89(C22×D5), (C2×C10)⋊9SD16, C101(C2×SD16), (C2×C4).99D20, C4.44(C2×D20), (C22×C8)⋊10D5, C51(C22×SD16), (C22×C40)⋊14C2, (C2×C40)⋊46C22, (C2×C20).390D4, C20.289(C2×D4), C4.51(C23×D5), C22.69(C2×D20), C10.21(C22×D4), C2.23(C22×D20), (C2×C20).786C23, (C22×D20).10C2, (C22×C4).442D10, (C22×C10).144D4, (C22×Dic10)⋊11C2, (C2×Dic10)⋊56C22, (C2×D20).236C22, (C22×C20).525C22, (C2×C10).177(C2×D4), (C2×C4).735(C22×D5), SmallGroup(320,1411)

Series: Derived Chief Lower central Upper central

C1C20 — C22×C40⋊C2
C1C5C10C20D20C2×D20C22×D20 — C22×C40⋊C2
C5C10C20 — C22×C40⋊C2
C1C23C22×C4C22×C8

Generators and relations for C22×C40⋊C2
 G = < a,b,c,d | a2=b2=c40=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c19 >

Subgroups: 1342 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C5, C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×10], Q8 [×10], C23, C23 [×10], D5 [×4], C10, C10 [×6], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×9], C2×Q8 [×9], C24, Dic5 [×4], C20, C20 [×3], D10 [×16], C2×C10 [×7], C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C40 [×4], Dic10 [×4], Dic10 [×6], D20 [×4], D20 [×6], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×10], C22×C10, C22×SD16, C40⋊C2 [×16], C2×C40 [×6], C2×Dic10 [×6], C2×Dic10 [×3], C2×D20 [×6], C2×D20 [×3], C22×Dic5, C22×C20, C23×D5, C2×C40⋊C2 [×12], C22×C40, C22×Dic10, C22×D20, C22×C40⋊C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, SD16 [×4], C2×D4 [×6], C24, D10 [×7], C2×SD16 [×6], C22×D4, D20 [×4], C22×D5 [×7], C22×SD16, C40⋊C2 [×4], C2×D20 [×6], C23×D5, C2×C40⋊C2 [×6], C22×D20, C22×C40⋊C2

Smallest permutation representation of C22×C40⋊C2
On 160 points
Generators in S160
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 41)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 145)(104 146)(105 147)(106 148)(107 149)(108 150)(109 151)(110 152)(111 153)(112 154)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 121)(120 122)
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 139)(42 140)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 159)(62 160)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 63)(43 61)(44 80)(45 59)(46 78)(47 57)(48 76)(49 55)(50 74)(51 53)(52 72)(54 70)(56 68)(58 66)(60 64)(65 79)(67 77)(69 75)(71 73)(81 95)(82 114)(83 93)(84 112)(85 91)(86 110)(87 89)(88 108)(90 106)(92 104)(94 102)(96 100)(97 119)(99 117)(101 115)(103 113)(105 111)(107 109)(116 120)(121 139)(122 158)(123 137)(124 156)(125 135)(126 154)(127 133)(128 152)(129 131)(130 150)(132 148)(134 146)(136 144)(138 142)(141 159)(143 157)(145 155)(147 153)(149 151)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H5A5B8A···8H10A···10N20A···20P40A···40AF
order12···2222244444444558···810···1020···2040···40
size11···120202020222220202020222···22···22···22···2

92 irreducible representations

dim11111222222222
type++++++++++++
imageC1C2C2C2C2D4D4D5SD16D10D10D20D20C40⋊C2
kernelC22×C40⋊C2C2×C40⋊C2C22×C40C22×Dic10C22×D20C2×C20C22×C10C22×C8C2×C10C2×C8C22×C4C2×C4C23C22
# reps112111312812212432

Matrix representation of C22×C40⋊C2 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
1000
04000
00400
00040
,
40000
0100
003124
001314
,
40000
04000
00636
00735
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,31,13,0,0,24,14],[40,0,0,0,0,40,0,0,0,0,6,7,0,0,36,35] >;

C22×C40⋊C2 in GAP, Magma, Sage, TeX

C_2^2\times C_{40}\rtimes C_2
% in TeX

G:=Group("C2^2xC40:C2");
// GroupNames label

G:=SmallGroup(320,1411);
// by ID

G=gap.SmallGroup(320,1411);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,80,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^40=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^19>;
// generators/relations

׿
×
𝔽