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G = C2×D20.3C4order 320 = 26·5

Direct product of C2 and D20.3C4

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

Aliases: C2×D20.3C4, C40.71C23, C20.67C24, (C2×C8)⋊37D10, C103(C8○D4), (C22×C8)⋊13D5, (C22×C40)⋊19C2, (C2×C40)⋊49C22, C4○D20.11C4, (C2×D20).31C4, D20.44(C2×C4), (C8×D5)⋊19C22, C23.39(C4×D5), C8.65(C22×D5), C4.66(C23×D5), C8⋊D521C22, C10.51(C23×C4), C52C8.31C23, (C4×D5).70C23, (C2×C20).880C23, C20.180(C22×C4), Dic10.47(C2×C4), (C2×Dic10).32C4, C4○D20.58C22, D10.21(C22×C4), (C22×C4).441D10, C4.Dic539C22, Dic5.20(C22×C4), (C22×C20).544C22, C54(C2×C8○D4), (D5×C2×C8)⋊25C2, C4.121(C2×C4×D5), C5⋊D4.9(C2×C4), (C2×C8⋊D5)⋊29C2, C22.11(C2×C4×D5), C2.31(D5×C22×C4), (C4×D5).59(C2×C4), (C2×C4).119(C4×D5), (C2×C5⋊D4).28C4, (C2×C20).409(C2×C4), (C2×C4○D20).29C2, (C2×C4.Dic5)⋊33C2, (C2×C4×D5).385C22, (C22×D5).82(C2×C4), (C2×C4).824(C22×D5), (C2×C10).257(C22×C4), (C22×C10).171(C2×C4), (C2×C52C8).335C22, (C2×Dic5).116(C2×C4), SmallGroup(320,1410)

Series: Derived Chief Lower central Upper central

C1C10 — C2×D20.3C4
C1C5C10C20C4×D5C2×C4×D5C2×C4○D20 — C2×D20.3C4
C5C10 — C2×D20.3C4
C1C2×C8C22×C8

Generators and relations for C2×D20.3C4
 G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 718 in 266 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×10], M4(2) [×12], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8, C22×C8 [×2], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C52C8 [×4], C40 [×4], Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C2×C8○D4, C8×D5 [×8], C8⋊D5 [×8], C2×C52C8 [×2], C4.Dic5 [×4], C2×C40 [×2], C2×C40 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, D5×C2×C8 [×2], C2×C8⋊D5 [×2], D20.3C4 [×8], C2×C4.Dic5, C22×C40, C2×C4○D20, C2×D20.3C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C8○D4 [×2], C23×C4, C4×D5 [×4], C22×D5 [×7], C2×C8○D4, C2×C4×D5 [×6], C23×D5, D20.3C4 [×2], D5×C22×C4, C2×D20.3C4

Smallest permutation representation of C2×D20.3C4
On 160 points
Generators in S160
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 131)(39 132)(40 133)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 141)(77 142)(78 143)(79 144)(80 145)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 101)(97 102)(98 103)(99 104)(100 105)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 60)(18 59)(19 58)(20 57)(21 133)(22 132)(23 131)(24 130)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 140)(35 139)(36 138)(37 137)(38 136)(39 135)(40 134)(61 159)(62 158)(63 157)(64 156)(65 155)(66 154)(67 153)(68 152)(69 151)(70 150)(71 149)(72 148)(73 147)(74 146)(75 145)(76 144)(77 143)(78 142)(79 141)(80 160)(81 107)(82 106)(83 105)(84 104)(85 103)(86 102)(87 101)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)
(1 87 73 36 11 97 63 26)(2 88 74 37 12 98 64 27)(3 89 75 38 13 99 65 28)(4 90 76 39 14 100 66 29)(5 91 77 40 15 81 67 30)(6 92 78 21 16 82 68 31)(7 93 79 22 17 83 69 32)(8 94 80 23 18 84 70 33)(9 95 61 24 19 85 71 34)(10 96 62 25 20 86 72 35)(41 106 152 123 51 116 142 133)(42 107 153 124 52 117 143 134)(43 108 154 125 53 118 144 135)(44 109 155 126 54 119 145 136)(45 110 156 127 55 120 146 137)(46 111 157 128 56 101 147 138)(47 112 158 129 57 102 148 139)(48 113 159 130 58 103 149 140)(49 114 160 131 59 104 150 121)(50 115 141 132 60 105 151 122)

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

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

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

104 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B8A···8H8I8J8K8L8M···8T10A···10N20A···20P40A···40AF
order12222222224444444444558···888888···810···1020···2040···40
size1111221010101011112210101010221···1222210···102···22···22···2

104 irreducible representations

dim111111111112222222
type++++++++++
imageC1C2C2C2C2C2C2C4C4C4C4D5D10D10C8○D4C4×D5C4×D5D20.3C4
kernelC2×D20.3C4D5×C2×C8C2×C8⋊D5D20.3C4C2×C4.Dic5C22×C40C2×C4○D20C2×Dic10C2×D20C4○D20C2×C5⋊D4C22×C8C2×C8C22×C4C10C2×C4C23C2
# reps122811122842122812432

Matrix representation of C2×D20.3C4 in GL5(𝔽41)

400000
040000
004000
00010
00001
,
10000
093600
003200
000640
00010
,
10000
093600
0163200
000640
0003535
,
320000
014000
001400
000320
000032

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,36,32,0,0,0,0,0,6,1,0,0,0,40,0],[1,0,0,0,0,0,9,16,0,0,0,36,32,0,0,0,0,0,6,35,0,0,0,40,35],[32,0,0,0,0,0,14,0,0,0,0,0,14,0,0,0,0,0,32,0,0,0,0,0,32] >;

C2×D20.3C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}._3C_4
% in TeX

G:=Group("C2xD20.3C4");
// GroupNames label

G:=SmallGroup(320,1410);
// by ID

G=gap.SmallGroup(320,1410);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,80,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

׿
×
𝔽