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G = C406C4⋊C2order 320 = 26·5

13rd semidirect product of C406C4 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C406C413C2, D4⋊C412D5, C4⋊C4.137D10, (C2×D4).27D10, C10.D87C2, C202D4.5C2, (C2×C8).116D10, D4⋊Dic59C2, D101C811C2, C4.53(C4○D20), C10.41(C4○D8), (C22×D5).19D4, C22.177(D4×D5), C20.151(C4○D4), C2.15(D8⋊D5), C4.80(D42D5), C10.33(C8⋊C22), (C2×C40).127C22, (C2×C20).219C23, (C2×Dic5).197D4, (D4×C10).40C22, C52(C23.19D4), C4⋊Dic5.73C22, C2.11(SD163D5), C2.14(D10.12D4), C10.22(C22.D4), C4⋊C47D53C2, (C5×D4⋊C4)⋊12C2, (C2×C4×D5).15C22, (C2×C10).232(C2×D4), (C5×C4⋊C4).20C22, (C2×C52C8).17C22, (C2×C4).326(C22×D5), SmallGroup(320,406)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C406C4⋊C2
C1C5C10C2×C10C2×C20C2×C4×D5C4⋊C47D5 — C406C4⋊C2
C5C10C2×C20 — C406C4⋊C2
C1C22C2×C4D4⋊C4

Generators and relations for C406C4⋊C2
 G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a31b2, cbc=b-1 >

Subgroups: 446 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C23.19D4, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C5⋊D4, D4×C10, C10.D8, C406C4, D101C8, D4⋊Dic5, C5×D4⋊C4, C4⋊C47D5, C202D4, C406C4⋊C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8⋊C22, C22×D5, C23.19D4, C4○D20, D4×D5, D42D5, D10.12D4, D8⋊D5, SD163D5, C406C4⋊C2

Smallest permutation representation of C406C4⋊C2
On 160 points
Generators in S160
(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 74 124 105)(2 53 125 84)(3 72 126 103)(4 51 127 82)(5 70 128 101)(6 49 129 120)(7 68 130 99)(8 47 131 118)(9 66 132 97)(10 45 133 116)(11 64 134 95)(12 43 135 114)(13 62 136 93)(14 41 137 112)(15 60 138 91)(16 79 139 110)(17 58 140 89)(18 77 141 108)(19 56 142 87)(20 75 143 106)(21 54 144 85)(22 73 145 104)(23 52 146 83)(24 71 147 102)(25 50 148 81)(26 69 149 100)(27 48 150 119)(28 67 151 98)(29 46 152 117)(30 65 153 96)(31 44 154 115)(32 63 155 94)(33 42 156 113)(34 61 157 92)(35 80 158 111)(36 59 159 90)(37 78 160 109)(38 57 121 88)(39 76 122 107)(40 55 123 86)
(1 74)(2 96)(3 56)(4 118)(5 78)(6 100)(7 60)(8 82)(9 42)(10 104)(11 64)(12 86)(13 46)(14 108)(15 68)(16 90)(17 50)(18 112)(19 72)(20 94)(21 54)(22 116)(23 76)(24 98)(25 58)(26 120)(27 80)(28 102)(29 62)(30 84)(31 44)(32 106)(33 66)(34 88)(35 48)(36 110)(37 70)(38 92)(39 52)(40 114)(41 141)(43 123)(45 145)(47 127)(49 149)(51 131)(53 153)(55 135)(57 157)(59 139)(61 121)(63 143)(65 125)(67 147)(69 129)(71 151)(73 133)(75 155)(77 137)(79 159)(81 140)(83 122)(85 144)(87 126)(89 148)(91 130)(93 152)(95 134)(97 156)(99 138)(101 160)(103 142)(105 124)(107 146)(109 128)(111 150)(113 132)(115 154)(117 136)(119 158)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444455888810···1010101010202020202020202040···40
size111182022441010202040224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C4○D20C8⋊C22D42D5D4×D5D8⋊D5SD163D5
kernelC406C4⋊C2C10.D8C406C4D101C8D4⋊Dic5C5×D4⋊C4C4⋊C47D5C202D4C2×Dic5C22×D5D4⋊C4C20C4⋊C4C2×C8C2×D4C10C4C10C4C22C2C2
# reps1111111111242224812244

Matrix representation of C406C4⋊C2 in GL4(𝔽41) generated by

14000
273800
00916
003614
,
203600
312100
003823
00373
,
21500
352000
002435
00717
G:=sub<GL(4,GF(41))| [14,27,0,0,0,38,0,0,0,0,9,36,0,0,16,14],[20,31,0,0,36,21,0,0,0,0,38,37,0,0,23,3],[21,35,0,0,5,20,0,0,0,0,24,7,0,0,35,17] >;

C406C4⋊C2 in GAP, Magma, Sage, TeX

C_{40}\rtimes_6C_4\rtimes C_2
% in TeX

G:=Group("C40:6C4:C2");
// GroupNames label

G:=SmallGroup(320,406);
// by ID

G=gap.SmallGroup(320,406);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^31*b^2,c*b*c=b^-1>;
// generators/relations

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