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G = (C2×D20)⋊22C4order 320 = 26·5

7th semidirect product of C2×D20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D20)⋊22C4, C10.89(C4×D4), C205(C22⋊C4), (C2×Dic5)⋊11D4, (C2×C4).144D20, (C2×C20).142D4, C41(D10⋊C4), C2.3(C20⋊D4), C2.5(C4⋊D20), C22.111(D4×D5), C22.48(C2×D20), C10.54(C4⋊D4), C10.15(C41D4), (C22×D20).12C2, (C22×C4).335D10, C2.18(D208C4), C2.2(C20.23D4), C10.51(C4.4D4), C54(C24.3C22), (C23×D5).18C22, C23.296(C22×D5), (C22×C10).352C23, (C22×C20).144C22, C22.28(Q82D5), (C22×Dic5).215C22, (C2×C4⋊C4)⋊6D5, (C10×C4⋊C4)⋊6C2, (C2×C4×Dic5)⋊1C2, (C2×C4).79(C4×D5), C22.137(C2×C4×D5), (C2×C20).259(C2×C4), (C2×C10).452(C2×D4), C10.84(C2×C22⋊C4), (C2×D10⋊C4)⋊10C2, C22.67(C2×C5⋊D4), C2.16(C2×D10⋊C4), (C2×C4).129(C5⋊D4), (C22×D5).26(C2×C4), (C2×C10).189(C4○D4), (C2×C10).221(C22×C4), SmallGroup(320,615)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×D20)⋊22C4
C1C5C10C2×C10C22×C10C23×D5C22×D20 — (C2×D20)⋊22C4
C5C2×C10 — (C2×D20)⋊22C4
C1C23C2×C4⋊C4

Generators and relations for (C2×D20)⋊22C4
 G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, dcd-1=ac=ca, ad=da, cbc=b-1, dbd-1=b11 >

Subgroups: 1182 in 258 conjugacy classes, 83 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C5, C2×C4 [×6], C2×C4 [×14], D4 [×8], C23, C23 [×16], D5 [×4], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C24 [×2], Dic5 [×4], C20 [×4], C20 [×2], D10 [×20], C2×C10 [×3], C2×C10 [×4], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C2×C20 [×6], C22×D5 [×4], C22×D5 [×12], C22×C10, C24.3C22, C4×Dic5 [×2], D10⋊C4 [×8], C5×C4⋊C4 [×2], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5 [×2], C2×C4×Dic5, C2×D10⋊C4 [×4], C10×C4⋊C4, C22×D20, (C2×D20)⋊22C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C24.3C22, D10⋊C4 [×4], C2×C4×D5, C2×D20, D4×D5 [×2], Q82D5 [×2], C2×C5⋊D4, D208C4 [×2], C4⋊D20 [×2], C2×D10⋊C4, C20⋊D4, C20.23D4, (C2×D20)⋊22C4

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I···4P5A5B10A···10N20A···20X
order12···22222444444444···45510···1020···20
size11···1202020202222444410···10222···24···4

68 irreducible representations

dim1111112222222244
type++++++++++++
imageC1C2C2C2C2C4D4D4D5C4○D4D10C4×D5D20C5⋊D4D4×D5Q82D5
kernel(C2×D20)⋊22C4C2×C4×Dic5C2×D10⋊C4C10×C4⋊C4C22×D20C2×D20C2×Dic5C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C2×C4C22C22
# reps1141184424688844

Matrix representation of (C2×D20)⋊22C4 in GL6(𝔽41)

4000000
0400000
0040000
0004000
000010
000001
,
3510000
4000000
007100
0040000
000001
0000400
,
3510000
660000
00344000
007700
0000400
000001
,
2130000
28390000
0024100
00401700
000090
0000032

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,7,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[35,6,0,0,0,0,1,6,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[2,28,0,0,0,0,13,39,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;

(C2×D20)⋊22C4 in GAP, Magma, Sage, TeX

(C_2\times D_{20})\rtimes_{22}C_4
% in TeX

G:=Group("(C2xD20):22C4");
// GroupNames label

G:=SmallGroup(320,615);
// by ID

G=gap.SmallGroup(320,615);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,422,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11>;
// generators/relations

׿
×
𝔽