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G = (C2×C4)⋊9D20order 320 = 26·5

1st semidirect product of C2×C4 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C4)⋊9D20, (C2×C20)⋊19D4, C2.5(C4×D20), (C2×D20)⋊16C4, C10.32(C4×D4), C10.2C22≀C2, (C2×Dic5)⋊15D4, C22.59(D4×D5), D102(C22⋊C4), C10.3(C4⋊D4), C2.1(C4⋊D20), (C22×D20).1C2, C2.C427D5, (C22×C4).15D10, C22.24(C2×D20), C2.1(C22⋊D20), C2.2(D10⋊D4), C2.5(D208C4), (C22×D5).103D4, C52(C23.23D4), C22.34(C4○D20), (C23×D5).94C22, C23.255(C22×D5), C10.10C4227C2, C2.3(D10.13D4), (C22×C10).290C23, (C22×C20).331C22, C22.17(Q82D5), C10.38(C22.D4), (C22×Dic5).14C22, (C2×C4)⋊2(C4×D5), (C2×C20)⋊17(C2×C4), (D5×C22×C4)⋊12C2, C2.7(D5×C22⋊C4), C22.89(C2×C4×D5), (C22×D5)⋊5(C2×C4), (C2×C10).199(C2×D4), C10.45(C2×C22⋊C4), (C2×D10⋊C4)⋊28C2, (C2×C10).183(C4○D4), (C5×C2.C42)⋊14C2, (C2×C10).150(C22×C4), SmallGroup(320,292)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×C4)⋊9D20
C1C5C10C2×C10C22×C10C23×D5C22×D20 — (C2×C4)⋊9D20
C5C2×C10 — (C2×C4)⋊9D20
C1C23C2.C42

Generators and relations for (C2×C4)⋊9D20
 G = < a,b,c,d | a2=b4=c20=d2=1, cbc-1=dbd=ab=ba, ac=ca, ad=da, dcd=c-1 >

Subgroups: 1294 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×6], C4 [×8], C22 [×7], C22 [×26], C5, C2×C4 [×4], C2×C4 [×22], D4 [×8], C23, C23 [×18], D5 [×6], C10 [×7], C22⋊C4 [×6], C22×C4 [×3], C22×C4 [×8], C2×D4 [×8], C24 [×2], Dic5 [×3], C20 [×5], D10 [×4], D10 [×22], C2×C10 [×7], C2.C42, C2.C42, C2×C22⋊C4 [×3], C23×C4, C22×D4, C4×D5 [×8], D20 [×8], C2×Dic5 [×2], C2×Dic5 [×5], C2×C20 [×4], C2×C20 [×7], C22×D5 [×8], C22×D5 [×10], C22×C10, C23.23D4, D10⋊C4 [×6], C2×C4×D5 [×6], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], C10.10C42, C5×C2.C42, C2×D10⋊C4 [×3], D5×C22×C4, C22×D20, (C2×C4)⋊9D20
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], C22≀C2, C4⋊D4 [×2], C22.D4, C4×D5 [×2], D20 [×2], C22×D5, C23.23D4, C2×C4×D5, C2×D20, C4○D20, D4×D5 [×3], Q82D5, C4×D20, D5×C22⋊C4, C22⋊D20, D10⋊D4, D208C4, D10.13D4, C4⋊D20, (C2×C4)⋊9D20

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B10A···10N20A···20X
order12···2222222444444444444445510···1020···20
size11···110101010202022224444101010102020222···24···4

68 irreducible representations

dim111111122222222244
type++++++++++++++
imageC1C2C2C2C2C2C4D4D4D4D5C4○D4D10C4×D5D20C4○D20D4×D5Q82D5
kernel(C2×C4)⋊9D20C10.10C42C5×C2.C42C2×D10⋊C4D5×C22×C4C22×D20C2×D20C2×Dic5C2×C20C22×D5C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22
# reps111311822424688862

Matrix representation of (C2×C4)⋊9D20 in GL6(𝔽41)

100000
010000
001000
000100
0000400
0000040
,
900000
090000
0032000
0003200
000001
000010
,
9110000
30140000
00344000
008100
000010
0000040
,
0400000
4000000
00404000
000100
000010
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,30,0,0,0,0,11,14,0,0,0,0,0,0,34,8,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

(C2×C4)⋊9D20 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_9D_{20}
% in TeX

G:=Group("(C2xC4):9D20");
// GroupNames label

G:=SmallGroup(320,292);
// by ID

G=gap.SmallGroup(320,292);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,12550]);
// Polycyclic

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

׿
×
𝔽