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G = C42.48D10order 320 = 26·5

48th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.48D10, D4⋊D59C4, D44(C4×D5), (C4×D4)⋊2D5, (D4×C20)⋊2C2, (C4×D20)⋊20C2, D2021(C2×C4), C55(D8⋊C4), C4⋊C4.242D10, C10.100(C4×D4), (C2×C20).254D4, D206C429C2, (C2×D4).189D10, C20.49(C4○D4), C4.37(C4○D20), D4⋊Dic511C2, C20.57(C22×C4), (C4×C20).85C22, C20.Q832C2, C42.D55C2, C2.3(D4⋊D10), (C2×C20).336C23, C10.108(C8⋊C22), C2.3(D4.D10), (C2×D20).245C22, (D4×C10).231C22, C4⋊Dic5.327C22, C4.22(C2×C4×D5), C52C88(C2×C4), (C5×D4)⋊19(C2×C4), (C2×D4⋊D5).4C2, C2.16(C4×C5⋊D4), (C2×C10).467(C2×D4), C22.76(C2×C5⋊D4), (C2×C4).217(C5⋊D4), (C5×C4⋊C4).273C22, (C2×C52C8).92C22, (C2×C4).436(C22×D5), SmallGroup(320,641)

Series: Derived Chief Lower central Upper central

C1C20 — C42.48D10
C1C5C10C2×C10C2×C20C2×D20C2×D4⋊D5 — C42.48D10
C5C10C20 — C42.48D10
C1C22C42C4×D4

Generators and relations for C42.48D10
 G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 502 in 132 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×2], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4, C2×D4, Dic5, C20 [×2], C20 [×3], D10 [×4], C2×C10, C2×C10 [×4], C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4, C4×D4, C2×D8, C52C8 [×2], C52C8, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×D5, C22×C10, D8⋊C4, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×4], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×C20, D4×C10, C42.D5, C20.Q8, D206C4, D4⋊Dic5, C4×D20, C2×D4⋊D5, D4×C20, C42.48D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, D8⋊C4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, D4.D10, D4⋊D10, C42.48D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G4H4I4J5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order122222224···4444455888810···1010···1020···2020···20
size11114420202···244202022202020202···24···42···24···4

62 irreducible representations

dim111111111222222222444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10D10C5⋊D4C4×D5C4○D20C8⋊C22D4.D10D4⋊D10
kernelC42.48D10C42.D5C20.Q8D206C4D4⋊Dic5C4×D20C2×D4⋊D5D4×C20D4⋊D5C2×C20C4×D4C20C42C4⋊C4C2×D4C2×C4D4C4C10C2C2
# reps111111118222222888244

Matrix representation of C42.48D10 in GL6(𝔽41)

900000
090000
0000241
00004017
00174000
0012400
,
4000000
0400000
000010
000001
0040000
0004000
,
1080000
3310000
00237723
0034151810
007231834
001810726
,
31330000
28100000
007231834
0015343123
00237723
0010181534

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,24,40,0,0,0,0,1,17,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[10,3,0,0,0,0,8,31,0,0,0,0,0,0,23,34,7,18,0,0,7,15,23,10,0,0,7,18,18,7,0,0,23,10,34,26],[31,28,0,0,0,0,33,10,0,0,0,0,0,0,7,15,23,10,0,0,23,34,7,18,0,0,18,31,7,15,0,0,34,23,23,34] >;

C42.48D10 in GAP, Magma, Sage, TeX

C_4^2._{48}D_{10}
% in TeX

G:=Group("C4^2.48D10");
// GroupNames label

G:=SmallGroup(320,641);
// by ID

G=gap.SmallGroup(320,641);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,1684,851,102,12550]);
// Polycyclic

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

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