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G = C42.282D10order 320 = 26·5

5th central extension by C42 of D10

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.282D10, C20.41M4(2), (C4×C8)⋊2D5, (C4×C40)⋊3C2, (C4×D5)⋊3C8, C4.22(C8×D5), C20.61(C2×C8), D10.8(C2×C8), (C2×C8).282D10, C4.16(C8⋊D5), C10.27(C22×C8), C20.8Q842C2, (C4×Dic5).15C4, Dic5.11(C2×C8), (D5×C42).11C2, D101C8.17C2, C20.240(C4○D4), C4.124(C4○D20), (C4×C20).338C22, (C2×C20).803C23, (C2×C40).341C22, C55(C42.12C4), C2.1(C42⋊D5), C10.36(C2×M4(2)), C10.26(C42⋊C2), (C4×Dic5).294C22, C2.5(D5×C2×C8), (C2×C4×D5).16C4, (C4×C52C8)⋊19C2, C2.1(C2×C8⋊D5), C22.35(C2×C4×D5), (C2×C4).173(C4×D5), (C2×C20).419(C2×C4), (C2×C4×D5).335C22, (C22×D5).94(C2×C4), (C2×C4).745(C22×D5), (C2×C10).159(C22×C4), (C2×C52C8).297C22, (C2×Dic5).134(C2×C4), SmallGroup(320,312)

Series: Derived Chief Lower central Upper central

C1C10 — C42.282D10
C1C5C10C20C2×C20C2×C4×D5D5×C42 — C42.282D10
C5C10 — C42.282D10
C1C42C4×C8

Generators and relations for C42.282D10
 G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 350 in 118 conjugacy classes, 63 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×4], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×2], C2×C8 [×2], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C4×C8, C4×C8, C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.12C4, C2×C52C8 [×2], C4×Dic5 [×3], C4×C20, C2×C40 [×2], C2×C4×D5 [×3], C4×C52C8, C20.8Q8 [×2], D101C8 [×2], C4×C40, D5×C42, C42.282D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C22×C8, C2×M4(2), C4×D5 [×2], C22×D5, C42.12C4, C8×D5 [×2], C8⋊D5 [×2], C2×C4×D5, C4○D20 [×2], C42⋊D5, D5×C2×C8, C2×C8⋊D5, C42.282D10

Smallest permutation representation of C42.282D10
On 160 points
Generators in S160
(1 109 55 126)(2 110 56 127)(3 111 57 128)(4 112 58 129)(5 113 59 130)(6 114 60 131)(7 115 61 132)(8 116 62 133)(9 117 63 134)(10 118 64 135)(11 119 65 136)(12 120 66 137)(13 81 67 138)(14 82 68 139)(15 83 69 140)(16 84 70 141)(17 85 71 142)(18 86 72 143)(19 87 73 144)(20 88 74 145)(21 89 75 146)(22 90 76 147)(23 91 77 148)(24 92 78 149)(25 93 79 150)(26 94 80 151)(27 95 41 152)(28 96 42 153)(29 97 43 154)(30 98 44 155)(31 99 45 156)(32 100 46 157)(33 101 47 158)(34 102 48 159)(35 103 49 160)(36 104 50 121)(37 105 51 122)(38 106 52 123)(39 107 53 124)(40 108 54 125)
(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 65 74 21 30 45 54)(2 73 66 29 22 53 46 9)(3 28 67 52 23 8 47 72)(4 51 68 7 24 71 48 27)(5 6 69 70 25 26 49 50)(11 20 75 44 31 40 55 64)(12 43 76 39 32 63 56 19)(13 38 77 62 33 18 57 42)(14 61 78 17 34 41 58 37)(15 16 79 80 35 36 59 60)(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)| (1,109,55,126)(2,110,56,127)(3,111,57,128)(4,112,58,129)(5,113,59,130)(6,114,60,131)(7,115,61,132)(8,116,62,133)(9,117,63,134)(10,118,64,135)(11,119,65,136)(12,120,66,137)(13,81,67,138)(14,82,68,139)(15,83,69,140)(16,84,70,141)(17,85,71,142)(18,86,72,143)(19,87,73,144)(20,88,74,145)(21,89,75,146)(22,90,76,147)(23,91,77,148)(24,92,78,149)(25,93,79,150)(26,94,80,151)(27,95,41,152)(28,96,42,153)(29,97,43,154)(30,98,44,155)(31,99,45,156)(32,100,46,157)(33,101,47,158)(34,102,48,159)(35,103,49,160)(36,104,50,121)(37,105,51,122)(38,106,52,123)(39,107,53,124)(40,108,54,125), (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), (1,10,65,74,21,30,45,54)(2,73,66,29,22,53,46,9)(3,28,67,52,23,8,47,72)(4,51,68,7,24,71,48,27)(5,6,69,70,25,26,49,50)(11,20,75,44,31,40,55,64)(12,43,76,39,32,63,56,19)(13,38,77,62,33,18,57,42)(14,61,78,17,34,41,58,37)(15,16,79,80,35,36,59,60)(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( (1,109,55,126)(2,110,56,127)(3,111,57,128)(4,112,58,129)(5,113,59,130)(6,114,60,131)(7,115,61,132)(8,116,62,133)(9,117,63,134)(10,118,64,135)(11,119,65,136)(12,120,66,137)(13,81,67,138)(14,82,68,139)(15,83,69,140)(16,84,70,141)(17,85,71,142)(18,86,72,143)(19,87,73,144)(20,88,74,145)(21,89,75,146)(22,90,76,147)(23,91,77,148)(24,92,78,149)(25,93,79,150)(26,94,80,151)(27,95,41,152)(28,96,42,153)(29,97,43,154)(30,98,44,155)(31,99,45,156)(32,100,46,157)(33,101,47,158)(34,102,48,159)(35,103,49,160)(36,104,50,121)(37,105,51,122)(38,106,52,123)(39,107,53,124)(40,108,54,125), (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), (1,10,65,74,21,30,45,54)(2,73,66,29,22,53,46,9)(3,28,67,52,23,8,47,72)(4,51,68,7,24,71,48,27)(5,6,69,70,25,26,49,50)(11,20,75,44,31,40,55,64)(12,43,76,39,32,63,56,19)(13,38,77,62,33,18,57,42)(14,61,78,17,34,41,58,37)(15,16,79,80,35,36,59,60)(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([(1,109,55,126),(2,110,56,127),(3,111,57,128),(4,112,58,129),(5,113,59,130),(6,114,60,131),(7,115,61,132),(8,116,62,133),(9,117,63,134),(10,118,64,135),(11,119,65,136),(12,120,66,137),(13,81,67,138),(14,82,68,139),(15,83,69,140),(16,84,70,141),(17,85,71,142),(18,86,72,143),(19,87,73,144),(20,88,74,145),(21,89,75,146),(22,90,76,147),(23,91,77,148),(24,92,78,149),(25,93,79,150),(26,94,80,151),(27,95,41,152),(28,96,42,153),(29,97,43,154),(30,98,44,155),(31,99,45,156),(32,100,46,157),(33,101,47,158),(34,102,48,159),(35,103,49,160),(36,104,50,121),(37,105,51,122),(38,106,52,123),(39,107,53,124),(40,108,54,125)], [(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)], [(1,10,65,74,21,30,45,54),(2,73,66,29,22,53,46,9),(3,28,67,52,23,8,47,72),(4,51,68,7,24,71,48,27),(5,6,69,70,25,26,49,50),(11,20,75,44,31,40,55,64),(12,43,76,39,32,63,56,19),(13,38,77,62,33,18,57,42),(14,61,78,17,34,41,58,37),(15,16,79,80,35,36,59,60),(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)])

104 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R5A5B8A···8H8I···8P10A···10F20A···20X40A···40AF
order1222224···44···4558···88···810···1020···2040···40
size111110101···110···10222···210···102···22···22···2

104 irreducible representations

dim111111111222222222
type+++++++++
imageC1C2C2C2C2C2C4C4C8D5M4(2)C4○D4D10D10C4×D5C8×D5C8⋊D5C4○D20
kernelC42.282D10C4×C52C8C20.8Q8D101C8C4×C40D5×C42C4×Dic5C2×C4×D5C4×D5C4×C8C20C20C42C2×C8C2×C4C4C4C4
# reps1122114416244248161616

Matrix representation of C42.282D10 in GL3(𝔽41) generated by

4000
0320
0032
,
900
010
001
,
3800
0615
0321
,
300
02015
01721
G:=sub<GL(3,GF(41))| [40,0,0,0,32,0,0,0,32],[9,0,0,0,1,0,0,0,1],[38,0,0,0,6,3,0,15,21],[3,0,0,0,20,17,0,15,21] >;

C42.282D10 in GAP, Magma, Sage, TeX

C_4^2._{282}D_{10}
% in TeX

G:=Group("C4^2.282D10");
// GroupNames label

G:=SmallGroup(320,312);
// by ID

G=gap.SmallGroup(320,312);
# by ID

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

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

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