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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 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×4], C23 [×2], D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8, C40, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C23.19D4, C2×C52C8, C4×Dic5, C4⋊Dic5 [×2], 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 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], 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)
(1 150 101 78)(2 129 102 57)(3 148 103 76)(4 127 104 55)(5 146 105 74)(6 125 106 53)(7 144 107 72)(8 123 108 51)(9 142 109 70)(10 121 110 49)(11 140 111 68)(12 159 112 47)(13 138 113 66)(14 157 114 45)(15 136 115 64)(16 155 116 43)(17 134 117 62)(18 153 118 41)(19 132 119 60)(20 151 120 79)(21 130 81 58)(22 149 82 77)(23 128 83 56)(24 147 84 75)(25 126 85 54)(26 145 86 73)(27 124 87 52)(28 143 88 71)(29 122 89 50)(30 141 90 69)(31 160 91 48)(32 139 92 67)(33 158 93 46)(34 137 94 65)(35 156 95 44)(36 135 96 63)(37 154 97 42)(38 133 98 61)(39 152 99 80)(40 131 100 59)
(1 150)(2 69)(3 132)(4 51)(5 154)(6 73)(7 136)(8 55)(9 158)(10 77)(11 140)(12 59)(13 122)(14 41)(15 144)(16 63)(17 126)(18 45)(19 148)(20 67)(21 130)(22 49)(23 152)(24 71)(25 134)(26 53)(27 156)(28 75)(29 138)(30 57)(31 160)(32 79)(33 142)(34 61)(35 124)(36 43)(37 146)(38 65)(39 128)(40 47)(42 105)(44 87)(46 109)(48 91)(50 113)(52 95)(54 117)(56 99)(58 81)(60 103)(62 85)(64 107)(66 89)(68 111)(70 93)(72 115)(74 97)(76 119)(78 101)(80 83)(82 121)(84 143)(86 125)(88 147)(90 129)(92 151)(94 133)(96 155)(98 137)(100 159)(102 141)(104 123)(106 145)(108 127)(110 149)(112 131)(114 153)(116 135)(118 157)(120 139)

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

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

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

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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