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G = C42.187D10order 320 = 26·5

7th non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.187D10, C203C87C2, C4⋊C4.6Dic5, C10.61(C8○D4), (C4×C20).19C22, C42⋊C2.8D5, C22⋊C4.3Dic5, C20.250(C4○D4), C4.134(C4○D20), C23.8(C2×Dic5), (C2×C20).847C23, C42.D522C2, (C22×C4).109D10, C2.5(D4.Dic5), C20.55D4.17C2, C57(C42.7C22), C10.65(C42⋊C2), (C22×C20).373C22, C22.44(C22×Dic5), C2.10(C23.21D10), (C4×C52C8)⋊26C2, (C5×C4⋊C4).22C4, (C2×C20).334(C2×C4), (C5×C22⋊C4).12C4, (C2×C4).19(C2×Dic5), (C5×C42⋊C2).9C2, (C2×C4).789(C22×D5), (C22×C10).126(C2×C4), (C2×C10).285(C22×C4), (C2×C52C8).322C22, SmallGroup(320,627)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.187D10
C1C5C10C20C2×C20C2×C52C8C4×C52C8 — C42.187D10
C5C2×C10 — C42.187D10
C1C2×C4C42⋊C2

Generators and relations for C42.187D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2c-1 >

Subgroups: 206 in 96 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], C23, C10, C10 [×2], C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C22×C4, C20 [×2], C20 [×5], C2×C10, C2×C10 [×3], C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C42⋊C2, C52C8 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×C10, C42.7C22, C2×C52C8 [×4], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C22×C20, C4×C52C8, C42.D5, C203C8 [×2], C20.55D4 [×2], C5×C42⋊C2, C42.187D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C42⋊C2, C8○D4 [×2], C2×Dic5 [×6], C22×D5, C42.7C22, C4○D20 [×2], C22×Dic5, C23.21D10, D4.Dic5 [×2], C42.187D10

Smallest permutation representation of C42.187D10
On 160 points
Generators in S160
(1 53 13 41)(2 59 14 47)(3 55 15 43)(4 51 11 49)(5 57 12 45)(6 54 19 42)(7 60 20 48)(8 56 16 44)(9 52 17 50)(10 58 18 46)(21 71 31 65)(22 77 32 61)(23 73 33 67)(24 79 34 63)(25 75 35 69)(26 76 36 70)(27 72 37 66)(28 78 38 62)(29 74 39 68)(30 80 40 64)(81 139 86 122)(82 128 87 135)(83 131 88 124)(84 130 89 137)(85 133 90 126)(91 136 96 129)(92 125 97 132)(93 138 98 121)(94 127 99 134)(95 140 100 123)(101 159 106 142)(102 148 107 155)(103 151 108 144)(104 150 109 157)(105 153 110 146)(111 156 116 149)(112 145 117 152)(113 158 118 141)(114 147 119 154)(115 160 120 143)
(1 28 10 23)(2 29 6 24)(3 30 7 25)(4 26 8 21)(5 27 9 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 62 46 67)(42 63 47 68)(43 64 48 69)(44 65 49 70)(45 66 50 61)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)(81 114 94 101)(82 115 95 102)(83 116 96 103)(84 117 97 104)(85 118 98 105)(86 119 99 106)(87 120 100 107)(88 111 91 108)(89 112 92 109)(90 113 93 110)(121 153 133 141)(122 154 134 142)(123 155 135 143)(124 156 136 144)(125 157 137 145)(126 158 138 146)(127 159 139 147)(128 160 140 148)(129 151 131 149)(130 152 132 150)
(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)
(1 116 38 91 10 103 33 88)(2 120 39 95 6 107 34 82)(3 114 40 99 7 101 35 86)(4 118 36 93 8 105 31 90)(5 112 37 97 9 109 32 84)(11 113 26 98 16 110 21 85)(12 117 27 92 17 104 22 89)(13 111 28 96 18 108 23 83)(14 115 29 100 19 102 24 87)(15 119 30 94 20 106 25 81)(41 144 78 131 46 156 73 129)(42 160 79 123 47 148 74 135)(43 142 80 139 48 154 75 127)(44 158 71 121 49 146 76 133)(45 150 72 137 50 152 77 125)(51 153 70 126 56 141 65 138)(52 145 61 132 57 157 66 130)(53 151 62 124 58 149 67 136)(54 143 63 140 59 155 68 128)(55 159 64 122 60 147 69 134)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K5A5B8A···8H8I8J8K8L10A···10F10G10H10I10J20A···20H20I···20AB
order1222244444444444558···8888810···101010101020···2020···20
size11114111122224442210···10202020202···244442···24···4

68 irreducible representations

dim11111111222222224
type++++++++--+
imageC1C2C2C2C2C2C4C4D5C4○D4D10Dic5Dic5D10C8○D4C4○D20D4.Dic5
kernelC42.187D10C4×C52C8C42.D5C203C8C20.55D4C5×C42⋊C2C5×C22⋊C4C5×C4⋊C4C42⋊C2C20C42C22⋊C4C4⋊C4C22×C4C10C4C2
# reps111221442444428168

Matrix representation of C42.187D10 in GL6(𝔽41)

900000
090000
001000
000100
00004023
0000321
,
4000000
0400000
0040000
0004000
0000320
0000032
,
100000
23400000
0037000
0071000
000010
0000940
,
23390000
18180000
00193800
00252200
00003828
000003

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,32,0,0,0,0,23,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,37,7,0,0,0,0,0,10,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[23,18,0,0,0,0,39,18,0,0,0,0,0,0,19,25,0,0,0,0,38,22,0,0,0,0,0,0,38,0,0,0,0,0,28,3] >;

C42.187D10 in GAP, Magma, Sage, TeX

C_4^2._{187}D_{10}
% in TeX

G:=Group("C4^2.187D10");
// GroupNames label

G:=SmallGroup(320,627);
// by ID

G=gap.SmallGroup(320,627);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,136,12550]);
// Polycyclic

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

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