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G = C42.70D10order 320 = 26·5

70th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.70D10, C4⋊C4.76D10, (C2×C20).85D4, C42.C22D5, C204D4.8C2, D206C441C2, C20.71(C4○D4), (C2×C20).385C23, (C4×C20).115C22, C4.13(Q82D5), C42.D511C2, C2.22(D4⋊D10), C10.123(C8⋊C22), C10.55(C4.4D4), C2.8(C20.23D4), (C2×D20).107C22, C53(C42.29C22), (C5×C42.C2)⋊2C2, (C2×C10).516(C2×D4), (C2×C4).67(C5⋊D4), (C5×C4⋊C4).123C22, (C2×C4).483(C22×D5), C22.189(C2×C5⋊D4), (C2×C52C8).127C22, SmallGroup(320,694)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.70D10
C1C5C10C20C2×C20C2×D20C204D4 — C42.70D10
C5C10C2×C20 — C42.70D10
C1C22C42C42.C2

Generators and relations for C42.70D10
 G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >

Subgroups: 590 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], D5 [×2], C10, C10 [×2], C42, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×D4 [×4], C20 [×2], C20 [×4], D10 [×6], C2×C10, C8⋊C4, D4⋊C4 [×4], C42.C2, C41D4, C52C8 [×2], D20 [×8], C2×C20, C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C42.29C22, C2×C52C8 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×D20 [×2], C2×D20 [×2], C42.D5, D206C4 [×4], C204D4, C5×C42.C2, C42.70D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C42.29C22, Q82D5 [×2], C2×C5⋊D4, C20.23D4, D4⋊D10 [×2], C42.70D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12222244444455888810···1020···2020···20
size1111404022448822202020202···24···48···8

44 irreducible representations

dim11111222222444
type++++++++++++
imageC1C2C2C2C2D4D5C4○D4D10D10C5⋊D4C8⋊C22Q82D5D4⋊D10
kernelC42.70D10C42.D5D206C4C204D4C5×C42.C2C2×C20C42.C2C20C42C4⋊C4C2×C4C10C4C2
# reps11411224248248

Matrix representation of C42.70D10 in GL6(𝔽41)

37240000
140000
0030322218
009112319
003032119
009113230
,
4000000
0400000
0010390
0001039
0010400
0001040
,
36120000
3250000
0066524
00355171
0029183535
002326636
,
36300000
3250000
0030111736
006114024
00122366
002129535

G:=sub<GL(6,GF(41))| [37,1,0,0,0,0,24,4,0,0,0,0,0,0,30,9,30,9,0,0,32,11,32,11,0,0,22,23,11,32,0,0,18,19,9,30],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,39,0,40,0,0,0,0,39,0,40],[36,32,0,0,0,0,12,5,0,0,0,0,0,0,6,35,29,23,0,0,6,5,18,26,0,0,5,17,35,6,0,0,24,1,35,36],[36,32,0,0,0,0,30,5,0,0,0,0,0,0,30,6,12,21,0,0,11,11,23,29,0,0,17,40,6,5,0,0,36,24,6,35] >;

C42.70D10 in GAP, Magma, Sage, TeX

C_4^2._{70}D_{10}
% in TeX

G:=Group("C4^2.70D10");
// GroupNames label

G:=SmallGroup(320,694);
// by ID

G=gap.SmallGroup(320,694);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,555,100,1123,297,136,12550]);
// Polycyclic

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

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