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G = C42.243D10order 320 = 26·5

2nd non-split extension by C42 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.243D10, (C4×C8)⋊4D5, (C4×C40)⋊4C2, (C2×C8).284D10, C20.8Q81C2, D101C8.1C2, C10.30(C8○D4), D10⋊C4.12C4, C20.243(C4○D4), C4.127(C4○D20), (C4×C20).341C22, (C2×C40).344C22, (C2×C20).808C23, C10.D4.12C4, C42⋊D5.10C2, C42.D516C2, C2.8(D20.3C4), C2.11(C42⋊D5), C53(C42.7C22), C10.27(C42⋊C2), (C4×Dic5).199C22, (C2×C4).89(C4×D5), C22.96(C2×C4×D5), (C2×C20).379(C2×C4), (C2×C4×D5).227C22, (C2×Dic5).14(C2×C4), (C22×D5).15(C2×C4), (C2×C4).750(C22×D5), (C2×C10).164(C22×C4), (C2×C52C8).192C22, SmallGroup(320,317)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.243D10
C1C5C10C20C2×C20C2×C4×D5C42⋊D5 — C42.243D10
C5C2×C10 — C42.243D10
C1C2×C4C4×C8

Generators and relations for C42.243D10
 G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 302 in 96 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×5], C23, D5, C10, C10 [×2], 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 [×2], C42⋊C2, C52C8 [×2], C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C42.7C22, C2×C52C8 [×2], C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C42.D5, C20.8Q8 [×2], D101C8 [×2], C4×C40, C42⋊D5, C42.243D10
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, C4○D20 [×2], C42⋊D5, D20.3C4 [×2], C42.243D10

Smallest permutation representation of C42.243D10
On 160 points
Generators in S160
(1 106 56 140)(2 107 57 141)(3 108 58 142)(4 109 59 143)(5 110 60 144)(6 111 61 145)(7 112 62 146)(8 113 63 147)(9 114 64 148)(10 115 65 149)(11 116 66 150)(12 117 67 151)(13 118 68 152)(14 119 69 153)(15 120 70 154)(16 81 71 155)(17 82 72 156)(18 83 73 157)(19 84 74 158)(20 85 75 159)(21 86 76 160)(22 87 77 121)(23 88 78 122)(24 89 79 123)(25 90 80 124)(26 91 41 125)(27 92 42 126)(28 93 43 127)(29 94 44 128)(30 95 45 129)(31 96 46 130)(32 97 47 131)(33 98 48 132)(34 99 49 133)(35 100 50 134)(36 101 51 135)(37 102 52 136)(38 103 53 137)(39 104 54 138)(40 105 55 139)
(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 66 75 21 30 46 55)(2 74 67 29 22 54 47 9)(3 28 68 53 23 8 48 73)(4 52 69 7 24 72 49 27)(5 6 70 71 25 26 50 51)(11 20 76 45 31 40 56 65)(12 44 77 39 32 64 57 19)(13 38 78 63 33 18 58 43)(14 62 79 17 34 42 59 37)(15 16 80 41 35 36 60 61)(81 144 125 120 101 124 145 100)(82 119 126 123 102 99 146 143)(83 122 127 98 103 142 147 118)(84 97 128 141 104 117 148 121)(85 140 129 116 105 160 149 96)(86 115 130 159 106 95 150 139)(87 158 131 94 107 138 151 114)(88 93 132 137 108 113 152 157)(89 136 133 112 109 156 153 92)(90 111 134 155 110 91 154 135)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K5A5B8A···8H8I8J8K8L10A···10F20A···20X40A···40AF
order1222244444444444558···8888810···1020···2040···40
size11112011112222202020222···2202020202···22···22···2

92 irreducible representations

dim1111111122222222
type+++++++++
imageC1C2C2C2C2C2C4C4D5C4○D4D10D10C8○D4C4×D5C4○D20D20.3C4
kernelC42.243D10C42.D5C20.8Q8D101C8C4×C40C42⋊D5C10.D4D10⋊C4C4×C8C20C42C2×C8C10C2×C4C4C2
# reps112211442424881632

Matrix representation of C42.243D10 in GL4(𝔽41) generated by

32000
03200
003226
0009
,
40000
04000
0090
0009
,
162500
16200
0035
00038
,
251600
21600
003836
00203
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,32,0,0,0,26,9],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[16,16,0,0,25,2,0,0,0,0,3,0,0,0,5,38],[25,2,0,0,16,16,0,0,0,0,38,20,0,0,36,3] >;

C42.243D10 in GAP, Magma, Sage, TeX

C_4^2._{243}D_{10}
% in TeX

G:=Group("C4^2.243D10");
// GroupNames label

G:=SmallGroup(320,317);
// by ID

G=gap.SmallGroup(320,317);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,422,58,136,12550]);
// 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,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations

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