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G = C2×C207D4order 320 = 26·5

Direct product of C2 and C207D4

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

Aliases: C2×C207D4, C234D20, C24.71D10, C2015(C2×D4), (C2×C20)⋊37D4, (C23×C4)⋊5D5, (C23×C20)⋊8C2, C222(C2×D20), C103(C4⋊D4), (C22×C4)⋊44D10, (C22×C10)⋊15D4, (C22×D20)⋊12C2, (C2×D20)⋊50C22, C4⋊Dic564C22, C2.33(C22×D20), (C2×C10).288C24, (C2×C20).705C23, (C22×C20)⋊60C22, C10.134(C22×D4), D10⋊C442C22, C22.83(C4○D20), (C23×D5).75C22, C22.303(C23×D5), C23.234(C22×D5), (C22×C10).417C23, (C23×C10).110C22, (C2×Dic5).150C23, (C22×D5).126C23, (C22×Dic5).162C22, C54(C2×C4⋊D4), C44(C2×C5⋊D4), (C2×C10)⋊11(C2×D4), (C2×C4)⋊16(C5⋊D4), (C2×C4⋊Dic5)⋊29C2, C2.71(C2×C4○D20), C10.63(C2×C4○D4), C2.7(C22×C5⋊D4), (C22×C5⋊D4)⋊11C2, (C2×C5⋊D4)⋊42C22, (C2×D10⋊C4)⋊14C2, (C2×C4).658(C22×D5), C22.104(C2×C5⋊D4), (C2×C10).114(C4○D4), SmallGroup(320,1462)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C207D4
C1C5C10C2×C10C22×D5C23×D5C22×D20 — C2×C207D4
C5C2×C10 — C2×C207D4
C1C23C23×C4

Generators and relations for C2×C207D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1646 in 426 conjugacy classes, 143 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×10], C22 [×32], C5, C2×C4 [×8], C2×C4 [×18], D4 [×24], C23, C23 [×6], C23 [×20], D5 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×24], C24, C24 [×2], Dic5 [×4], C20 [×4], C20 [×2], D10 [×20], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×8], C2×C20 [×10], C22×D5 [×4], C22×D5 [×12], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C4⋊D4, C4⋊Dic5 [×4], D10⋊C4 [×8], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×D5 [×2], C23×C10, C2×C4⋊Dic5, C2×D10⋊C4 [×2], C207D4 [×8], C22×D20, C22×C5⋊D4 [×2], C23×C20, C2×C207D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C4○D4 [×2], C24, D10 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, D20 [×4], C5⋊D4 [×4], C22×D5 [×7], C2×C4⋊D4, C2×D20 [×6], C4○D20 [×2], C2×C5⋊D4 [×6], C23×D5, C207D4 [×4], C22×D20, C2×C4○D20, C22×C5⋊D4, C2×C207D4

Smallest permutation representation of C2×C207D4
On 160 points
Generators in S160
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 41)(19 42)(20 43)(21 154)(22 155)(23 156)(24 157)(25 158)(26 159)(27 160)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 153)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)
(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)
(1 104 124 38)(2 103 125 37)(3 102 126 36)(4 101 127 35)(5 120 128 34)(6 119 129 33)(7 118 130 32)(8 117 131 31)(9 116 132 30)(10 115 133 29)(11 114 134 28)(12 113 135 27)(13 112 136 26)(14 111 137 25)(15 110 138 24)(16 109 139 23)(17 108 140 22)(18 107 121 21)(19 106 122 40)(20 105 123 39)(41 94 74 154)(42 93 75 153)(43 92 76 152)(44 91 77 151)(45 90 78 150)(46 89 79 149)(47 88 80 148)(48 87 61 147)(49 86 62 146)(50 85 63 145)(51 84 64 144)(52 83 65 143)(53 82 66 142)(54 81 67 141)(55 100 68 160)(56 99 69 159)(57 98 70 158)(58 97 71 157)(59 96 72 156)(60 95 73 155)
(1 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 158)(22 157)(23 156)(24 155)(25 154)(26 153)(27 152)(28 151)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 160)(40 159)(41 137)(42 136)(43 135)(44 134)(45 133)(46 132)(47 131)(48 130)(49 129)(50 128)(51 127)(52 126)(53 125)(54 124)(55 123)(56 122)(57 121)(58 140)(59 139)(60 138)(81 104)(82 103)(83 102)(84 101)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L5A5B10A···10AD20A···20AF
order12···2222222224···444445510···1020···20
size11···12222202020202···220202020222···22···2

92 irreducible representations

dim1111111222222222
type+++++++++++++
imageC1C2C2C2C2C2C2D4D4D5C4○D4D10D10C5⋊D4D20C4○D20
kernelC2×C207D4C2×C4⋊Dic5C2×D10⋊C4C207D4C22×D20C22×C5⋊D4C23×C20C2×C20C22×C10C23×C4C2×C10C22×C4C24C2×C4C23C22
# reps11281214424122161616

Matrix representation of C2×C207D4 in GL5(𝔽41)

400000
01000
00100
00010
00001
,
400000
040800
040700
000132
0003925
,
400000
0382000
020300
000400
000351
,
10000
073300
063400
000400
000351

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,40,0,0,0,8,7,0,0,0,0,0,13,39,0,0,0,2,25],[40,0,0,0,0,0,38,20,0,0,0,20,3,0,0,0,0,0,40,35,0,0,0,0,1],[1,0,0,0,0,0,7,6,0,0,0,33,34,0,0,0,0,0,40,35,0,0,0,0,1] >;

C2×C207D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_7D_4
% in TeX

G:=Group("C2xC20:7D4");
// GroupNames label

G:=SmallGroup(320,1462);
// by ID

G=gap.SmallGroup(320,1462);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,12550]);
// Polycyclic

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

׿
×
𝔽