Copied to
clipboard

G = C408C4⋊C2order 320 = 26·5

12nd semidirect product of C408C4 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C408C412C2, C22⋊C8.8D5, (C8×Dic5)⋊14C2, (C2×C8).193D10, C4⋊Dic5.22C4, C23.11(C4×D5), C10.31(C8○D4), C20.8Q817C2, (C22×C4).74D10, C23.D5.10C4, C20.296(C4○D4), (C2×C40).169C22, (C2×C20).818C23, C20.55D4.2C2, C4.122(D42D5), C2.9(D20.2C4), C2.9(D20.3C4), (C22×C20).91C22, C55(C42.7C22), C10.41(C42⋊C2), (C4×Dic5).301C22, C23.21D10.2C2, C2.10(C23.11D10), (C2×C4).30(C4×D5), C22.102(C2×C4×D5), (C2×C20).211(C2×C4), (C5×C22⋊C8).11C2, (C2×Dic5).18(C2×C4), (C2×C4).760(C22×D5), (C2×C10).174(C22×C4), (C22×C10).104(C2×C4), (C2×C52C8).307C22, SmallGroup(320,347)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C408C4⋊C2
C1C5C10C20C2×C20C4×Dic5C23.21D10 — C408C4⋊C2
C5C2×C10 — C408C4⋊C2
C1C2×C4C22⋊C8

Generators and relations for C408C4⋊C2
 G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=ab2, cbc=a20b >

Subgroups: 254 in 96 conjugacy classes, 47 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C23, C10 [×3], C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×3], C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8 [×2], C42⋊C2, C52C8 [×2], C40 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×C10, C42.7C22, C2×C52C8 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C2×C40 [×2], C22×C20, C8×Dic5, C20.8Q8 [×2], C408C4, C20.55D4, C5×C22⋊C8, C23.21D10, C408C4⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C8○D4 [×2], C4×D5 [×2], C22×D5, C42.7C22, C2×C4×D5, D42D5 [×2], C23.11D10, D20.3C4, D20.2C4, C408C4⋊C2

Smallest permutation representation of C408C4⋊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 59 91 142)(2 48 92 131)(3 77 93 160)(4 66 94 149)(5 55 95 138)(6 44 96 127)(7 73 97 156)(8 62 98 145)(9 51 99 134)(10 80 100 123)(11 69 101 152)(12 58 102 141)(13 47 103 130)(14 76 104 159)(15 65 105 148)(16 54 106 137)(17 43 107 126)(18 72 108 155)(19 61 109 144)(20 50 110 133)(21 79 111 122)(22 68 112 151)(23 57 113 140)(24 46 114 129)(25 75 115 158)(26 64 116 147)(27 53 117 136)(28 42 118 125)(29 71 119 154)(30 60 120 143)(31 49 81 132)(32 78 82 121)(33 67 83 150)(34 56 84 139)(35 45 85 128)(36 74 86 157)(37 63 87 146)(38 52 88 135)(39 41 89 124)(40 70 90 153)
(2 92)(4 94)(6 96)(8 98)(10 100)(12 102)(14 104)(16 106)(18 108)(20 110)(22 112)(24 114)(26 116)(28 118)(30 120)(32 82)(34 84)(36 86)(38 88)(40 90)(41 61)(42 145)(43 63)(44 147)(45 65)(46 149)(47 67)(48 151)(49 69)(50 153)(51 71)(52 155)(53 73)(54 157)(55 75)(56 159)(57 77)(58 121)(59 79)(60 123)(62 125)(64 127)(66 129)(68 131)(70 133)(72 135)(74 137)(76 139)(78 141)(80 143)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222444444444445588888888888810···101010101020···202020202040···40
size1111411114101010102020222222441010101020202···244442···244444···4

68 irreducible representations

dim1111111112222222244
type++++++++++-
imageC1C2C2C2C2C2C2C4C4D5C4○D4D10D10C8○D4C4×D5C4×D5D20.3C4D42D5D20.2C4
kernelC408C4⋊C2C8×Dic5C20.8Q8C408C4C20.55D4C5×C22⋊C8C23.21D10C4⋊Dic5C23.D5C22⋊C8C20C2×C8C22×C4C10C2×C4C23C2C4C2
# reps11211114424428441644

Matrix representation of C408C4⋊C2 in GL4(𝔽41) generated by

17000
241500
00920
003732
,
11700
04000
0090
0009
,
1000
244000
0010
003640
G:=sub<GL(4,GF(41))| [17,24,0,0,0,15,0,0,0,0,9,37,0,0,20,32],[1,0,0,0,17,40,0,0,0,0,9,0,0,0,0,9],[1,24,0,0,0,40,0,0,0,0,1,36,0,0,0,40] >;

C408C4⋊C2 in GAP, Magma, Sage, TeX

C_{40}\rtimes_8C_4\rtimes C_2
% in TeX

G:=Group("C40:8C4:C2");
// GroupNames label

G:=SmallGroup(320,347);
// by ID

G=gap.SmallGroup(320,347);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a*b^2,c*b*c=a^20*b>;
// generators/relations

׿
×
𝔽