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G = C2×C4.12D20order 320 = 26·5

Direct product of C2 and C4.12D20

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

Aliases: C2×C4.12D20, M4(2).32D10, (C2×C4).53D20, C4.67(C2×D20), C20.422(C2×D4), (C2×C20).176D4, C23.56(C4×D5), C102(C4.10D4), (C2×C20).418C23, (C2×Dic10).28C4, (C22×C4).144D10, (C2×M4(2)).17D5, C4.29(D10⋊C4), (C22×Dic5).5C4, C20.101(C22⋊C4), (C10×M4(2)).28C2, C4.Dic5.43C22, (C22×C20).191C22, (C22×Dic10).16C2, (C5×M4(2)).35C22, C22.51(D10⋊C4), (C2×Dic10).287C22, (C2×C4).55(C4×D5), C54(C2×C4.10D4), C22.22(C2×C4×D5), C4.115(C2×C5⋊D4), (C2×C20).284(C2×C4), (C2×Dic5).6(C2×C4), C2.33(C2×D10⋊C4), (C2×C4).257(C5⋊D4), C10.102(C2×C22⋊C4), (C2×C4).122(C22×D5), (C2×C4.Dic5).26C2, (C2×C10).117(C22×C4), (C22×C10).140(C2×C4), (C2×C10).131(C22⋊C4), SmallGroup(320,763)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C4.12D20
C1C5C10C20C2×C20C2×Dic10C22×Dic10 — C2×C4.12D20
C5C10C2×C10 — C2×C4.12D20
C1C22C22×C4C2×M4(2)

Generators and relations for C2×C4.12D20
 G = < a,b,c,d | a2=b20=1, c4=d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c3 >

Subgroups: 478 in 146 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×6], C2×C4 [×8], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×8], Dic5 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×2], C4.10D4 [×4], C2×M4(2), C2×M4(2), C22×Q8, C52C8 [×2], C40 [×2], Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C2×C4.10D4, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C4.12D20 [×4], C2×C4.Dic5, C10×M4(2), C22×Dic10, C2×C4.12D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4.10D4 [×2], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C4.10D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C4.12D20 [×2], C2×D10⋊C4, C2×C4.12D20

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444558888888810···101010101020···202020202040···40
size111122222220202020224444202020202···244442···244444···4

62 irreducible representations

dim11111112222222244
type++++++++++--
imageC1C2C2C2C2C4C4D4D5D10D10C4×D5D20C5⋊D4C4×D5C4.10D4C4.12D20
kernelC2×C4.12D20C4.12D20C2×C4.Dic5C10×M4(2)C22×Dic10C2×Dic10C22×Dic5C2×C20C2×M4(2)M4(2)C22×C4C2×C4C2×C4C2×C4C23C10C2
# reps14111444242488428

Matrix representation of C2×C4.12D20 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
1400000
3660000
0091100
00301400
00003911
00001425
,
3530000
1560000
00124439
0017402739
00817231
00109318
,
28270000
12130000
0003200
0032000
0000320
000049

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[35,15,0,0,0,0,3,6,0,0,0,0,0,0,1,17,8,10,0,0,24,40,17,9,0,0,4,27,23,3,0,0,39,39,1,18],[28,12,0,0,0,0,27,13,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,32,4,0,0,0,0,0,9] >;

C2×C4.12D20 in GAP, Magma, Sage, TeX

C_2\times C_4._{12}D_{20}
% in TeX

G:=Group("C2xC4.12D20");
// GroupNames label

G:=SmallGroup(320,763);
// by ID

G=gap.SmallGroup(320,763);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,58,1123,136,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^3>;
// generators/relations

׿
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