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G = C42.202D10order 320 = 26·5

22nd non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.202D10, D10.6M4(2), C20.24M4(2), C4⋊C811D5, C203C813C2, C408C421C2, (C2×C8).181D10, (D5×C42).3C2, D101C8.8C2, C4.11(C8⋊D5), (C4×C20).61C22, C57(C42.6C4), (C4×Dic5).22C4, C2.16(D5×M4(2)), C20.305(C4○D4), (C2×C20).832C23, (C2×C40).211C22, C4.54(Q82D5), C10.61(C2×M4(2)), C4.131(D42D5), C10.51(C42⋊C2), (C4×Dic5).307C22, (C5×C4⋊C8)⋊16C2, (C2×C4×D5).23C4, (C2×C4).146(C4×D5), C2.12(C2×C8⋊D5), C22.110(C2×C4×D5), (C2×C20).243(C2×C4), C2.9(C4⋊C47D5), (C2×C4×D5).348C22, (C2×C4).774(C22×D5), (C2×C10).188(C22×C4), (C2×C52C8).198C22, (C2×Dic5).144(C2×C4), (C22×D5).102(C2×C4), SmallGroup(320,462)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.202D10
C1C5C10C20C2×C20C2×C4×D5D5×C42 — C42.202D10
C5C2×C10 — C42.202D10
C1C2×C4C4⋊C8

Generators and relations for C42.202D10
 G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b-1, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c9 >

Subgroups: 350 in 110 conjugacy classes, 53 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×9], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×2], C2×C8 [×2], C22×C4 [×3], Dic5 [×3], C20 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8, C4⋊C8, C2×C42, C52C8 [×2], C40 [×2], C4×D5 [×6], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.6C4, C2×C52C8 [×2], C4×Dic5 [×3], C4×C20, C2×C40 [×2], C2×C4×D5 [×3], C203C8, C408C4 [×2], D101C8 [×2], C5×C4⋊C8, D5×C42, C42.202D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, M4(2) [×4], C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C2×M4(2) [×2], C4×D5 [×2], C22×D5, C42.6C4, C8⋊D5 [×2], C2×C4×D5, D42D5, Q82D5, C4⋊C47D5, C2×C8⋊D5, D5×M4(2), C42.202D10

Smallest permutation representation of C42.202D10
On 160 points
Generators in S160
(1 119 48 156)(2 157 49 120)(3 81 50 158)(4 159 51 82)(5 83 52 160)(6 121 53 84)(7 85 54 122)(8 123 55 86)(9 87 56 124)(10 125 57 88)(11 89 58 126)(12 127 59 90)(13 91 60 128)(14 129 61 92)(15 93 62 130)(16 131 63 94)(17 95 64 132)(18 133 65 96)(19 97 66 134)(20 135 67 98)(21 99 68 136)(22 137 69 100)(23 101 70 138)(24 139 71 102)(25 103 72 140)(26 141 73 104)(27 105 74 142)(28 143 75 106)(29 107 76 144)(30 145 77 108)(31 109 78 146)(32 147 79 110)(33 111 80 148)(34 149 41 112)(35 113 42 150)(36 151 43 114)(37 115 44 152)(38 153 45 116)(39 117 46 154)(40 155 47 118)
(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 78 47 21 30 58 67)(2 46 79 29 22 66 59 9)(3 28 80 65 23 8 60 45)(4 64 41 7 24 44 61 27)(5 6 42 43 25 26 62 63)(11 20 48 57 31 40 68 77)(12 56 49 39 32 76 69 19)(13 38 50 75 33 18 70 55)(14 74 51 17 34 54 71 37)(15 16 52 53 35 36 72 73)(81 106 148 133 101 86 128 153)(82 132 149 85 102 152 129 105)(83 84 150 151 103 104 130 131)(87 120 154 147 107 100 134 127)(88 146 155 99 108 126 135 119)(89 98 156 125 109 118 136 145)(90 124 157 117 110 144 137 97)(91 116 158 143 111 96 138 123)(92 142 159 95 112 122 139 115)(93 94 160 121 113 114 140 141)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H20I···20P40A···40P
order122222444444444···4558888888810···1020···2020···2040···40
size111110101111222210···10224444202020202···22···24···44···4

68 irreducible representations

dim1111111122222222444
type+++++++++-+
imageC1C2C2C2C2C2C4C4D5M4(2)C4○D4M4(2)D10D10C4×D5C8⋊D5D42D5Q82D5D5×M4(2)
kernelC42.202D10C203C8C408C4D101C8C5×C4⋊C8D5×C42C4×Dic5C2×C4×D5C4⋊C8C20C20D10C42C2×C8C2×C4C4C4C4C2
# reps11221144244424816224

Matrix representation of C42.202D10 in GL4(𝔽41) generated by

1000
0100
0090
003932
,
32000
03200
00400
00040
,
14000
1800
00940
00032
,
40100
8100
00321
0029
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,39,0,0,0,32],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[1,1,0,0,40,8,0,0,0,0,9,0,0,0,40,32],[40,8,0,0,1,1,0,0,0,0,32,2,0,0,1,9] >;

C42.202D10 in GAP, Magma, Sage, TeX

C_4^2._{202}D_{10}
% in TeX

G:=Group("C4^2.202D10");
// GroupNames label

G:=SmallGroup(320,462);
// by ID

G=gap.SmallGroup(320,462);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,1094,219,58,136,12550]);
// Polycyclic

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

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