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G = C42.82D10order 320 = 26·5

82nd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.82D10, C4⋊Q87D5, C4⋊C4.85D10, (C2×C20).296D4, C20.84(C4○D4), C10.Q1642C2, D206C4.16C2, C10.99(C8⋊C22), (C2×C20).407C23, (C4×C20).136C22, C4.17(Q82D5), C4.D20.11C2, C42.D515C2, C10.59(C4.4D4), (C2×D20).114C22, C10.98(C8.C22), C2.20(D4.D10), C2.19(C20.C23), C2.12(C20.23D4), C55(C42.28C22), (C2×Dic10).118C22, (C5×C4⋊Q8)⋊7C2, (C2×C10).538(C2×D4), (C2×C4).74(C5⋊D4), (C5×C4⋊C4).132C22, (C2×C4).504(C22×D5), C22.210(C2×C5⋊D4), (C2×C52C8).139C22, SmallGroup(320,716)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.82D10
C1C5C10C20C2×C20C2×D20C4.D20 — C42.82D10
C5C10C2×C20 — C42.82D10
C1C22C42C4⋊Q8

Generators and relations for C42.82D10
 G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >

Subgroups: 414 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×D4, C2×Q8 [×2], Dic5, C20 [×2], C20 [×4], D10 [×3], C2×C10, C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C52C8 [×2], Dic10 [×2], D20 [×2], C2×Dic5, C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C22×D5, C42.28C22, C2×C52C8 [×2], D10⋊C4 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×D20, Q8×C10, C42.D5, D206C4 [×2], C10.Q16 [×2], C4.D20, C5×C4⋊Q8, C42.82D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22, C8.C22, C5⋊D4 [×2], C22×D5, C42.28C22, Q82D5 [×2], C2×C5⋊D4, D4.D10, C20.C23, C20.23D4, C42.82D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12222444444455888810···1020···2020···20
size1111402244884022202020202···24···48···8

44 irreducible representations

dim11111122222244444
type+++++++++++-+
imageC1C2C2C2C2C2D4D5C4○D4D10D10C5⋊D4C8⋊C22C8.C22Q82D5D4.D10C20.C23
kernelC42.82D10C42.D5D206C4C10.Q16C4.D20C5×C4⋊Q8C2×C20C4⋊Q8C20C42C4⋊C4C2×C4C10C10C4C2C2
# reps11221122424811444

Matrix representation of C42.82D10 in GL8(𝔽41)

133335350000
828600000
0372180000
4440200000
000000185
000000123
0000233600
0000401800
,
400000000
040000000
004000000
000400000
00000010
00000001
000040000
000004000
,
3830210000
381723200000
91321380000
0372660000
0000427377
00003031263
00003773714
00002631110
,
3380200000
17382330000
13921240000
37026200000
00001027334
000021312938
00003871027
00001232131

G:=sub<GL(8,GF(41))| [13,8,0,4,0,0,0,0,33,28,37,4,0,0,0,0,35,6,21,40,0,0,0,0,35,0,8,20,0,0,0,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18,0,0,0,0,18,1,0,0,0,0,0,0,5,23,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[38,38,9,0,0,0,0,0,3,17,13,37,0,0,0,0,0,23,21,26,0,0,0,0,21,20,38,6,0,0,0,0,0,0,0,0,4,30,37,26,0,0,0,0,27,31,7,3,0,0,0,0,37,26,37,11,0,0,0,0,7,3,14,10],[3,17,13,37,0,0,0,0,38,38,9,0,0,0,0,0,0,23,21,26,0,0,0,0,20,3,24,20,0,0,0,0,0,0,0,0,10,21,38,12,0,0,0,0,27,31,7,3,0,0,0,0,3,29,10,21,0,0,0,0,34,38,27,31] >;

C42.82D10 in GAP, Magma, Sage, TeX

C_4^2._{82}D_{10}
% in TeX

G:=Group("C4^2.82D10");
// GroupNames label

G:=SmallGroup(320,716);
// by ID

G=gap.SmallGroup(320,716);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,555,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,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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