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G = C42.31D10order 320 = 26·5

31st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.31D10, C4⋊C816D5, C203C816C2, C408C422C2, (C8×Dic5)⋊24C2, (C2×C8).218D10, D101C8.9C2, C10.38(C8○D4), (C4×C20).66C22, C42⋊D5.2C2, D10⋊C4.25C4, C20.337(C4○D4), (C2×C20).837C23, (C2×C40).216C22, C4.57(Q82D5), C10.D4.25C4, C4.132(D42D5), C56(C42.7C22), C10.52(C42⋊C2), C2.16(D20.3C4), C2.15(D20.2C4), (C4×Dic5).309C22, (C5×C4⋊C8)⋊21C2, (C2×C4).37(C4×D5), C22.115(C2×C4×D5), (C2×C20).355(C2×C4), (C2×C4×D5).233C22, C2.10(C4⋊C47D5), (C22×D5).22(C2×C4), (C2×C4).779(C22×D5), (C2×C10).193(C22×C4), (C2×C52C8).201C22, (C2×Dic5).100(C2×C4), SmallGroup(320,467)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.31D10
C1C5C10C20C2×C20C2×C4×D5C42⋊D5 — C42.31D10
C5C2×C10 — C42.31D10
C1C2×C4C4⋊C8

Generators and relations for C42.31D10
 G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 302 in 96 conjugacy classes, 47 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×4], C2×C4 [×3], C2×C4 [×5], C23, D5, C10 [×3], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4⋊C8, C42⋊C2, C52C8 [×2], C40 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.7C22, C2×C52C8 [×2], C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C203C8, C8×Dic5, C408C4, D101C8 [×2], C5×C4⋊C8, C42⋊D5, C42.31D10
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, Q82D5, C4⋊C47D5, D20.3C4, D20.2C4, C42.31D10

Smallest permutation representation of C42.31D10
On 160 points
Generators in S160
(1 141 107 48)(2 49 108 142)(3 143 109 50)(4 51 110 144)(5 145 111 52)(6 53 112 146)(7 147 113 54)(8 55 114 148)(9 149 115 56)(10 57 116 150)(11 151 117 58)(12 59 118 152)(13 153 119 60)(14 61 120 154)(15 155 81 62)(16 63 82 156)(17 157 83 64)(18 65 84 158)(19 159 85 66)(20 67 86 160)(21 121 87 68)(22 69 88 122)(23 123 89 70)(24 71 90 124)(25 125 91 72)(26 73 92 126)(27 127 93 74)(28 75 94 128)(29 129 95 76)(30 77 96 130)(31 131 97 78)(32 79 98 132)(33 133 99 80)(34 41 100 134)(35 135 101 42)(36 43 102 136)(37 137 103 44)(38 45 104 138)(39 139 105 46)(40 47 106 140)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 10 117 86 21 30 97 106)(2 85 118 29 22 105 98 9)(3 28 119 104 23 8 99 84)(4 103 120 7 24 83 100 27)(5 6 81 82 25 26 101 102)(11 20 87 96 31 40 107 116)(12 95 88 39 32 115 108 19)(13 38 89 114 33 18 109 94)(14 113 90 17 34 93 110 37)(15 16 91 92 35 36 111 112)(41 147 144 64 61 127 124 44)(42 63 145 126 62 43 125 146)(45 143 148 60 65 123 128 80)(46 59 149 122 66 79 129 142)(47 121 150 78 67 141 130 58)(48 77 151 140 68 57 131 160)(49 139 152 56 69 159 132 76)(50 55 153 158 70 75 133 138)(51 157 154 74 71 137 134 54)(52 73 155 136 72 53 135 156)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F20A···20H20I···20P40A···40P
order12222444444444445588888888888810···1020···2020···2040···40
size1111201111441010101020222222441010101020202···22···24···44···4

68 irreducible representations

dim1111111112222222444
type++++++++++-+
imageC1C2C2C2C2C2C2C4C4D5C4○D4D10D10C8○D4C4×D5D20.3C4D42D5Q82D5D20.2C4
kernelC42.31D10C203C8C8×Dic5C408C4D101C8C5×C4⋊C8C42⋊D5C10.D4D10⋊C4C4⋊C8C20C42C2×C8C10C2×C4C2C4C4C2
# reps11112114424248816224

Matrix representation of C42.31D10 in GL4(𝔽41) generated by

244000
11700
0090
003832
,
32000
03200
00400
00040
,
271600
251600
00142
00427
,
162700
162500
002739
003614
G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,0,0,9,38,0,0,0,32],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[27,25,0,0,16,16,0,0,0,0,14,4,0,0,2,27],[16,16,0,0,27,25,0,0,0,0,27,36,0,0,39,14] >;

C42.31D10 in GAP, Magma, Sage, TeX

C_4^2._{31}D_{10}
% in TeX

G:=Group("C4^2.31D10");
// GroupNames label

G:=SmallGroup(320,467);
// by ID

G=gap.SmallGroup(320,467);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,422,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations

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