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G = C40.31D4order 320 = 26·5

31st non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.31D4, C408C45C2, C4.23(D4×D5), C52C8.9D4, (C2×C8).89D10, C54(C8.2D4), C8.3(C5⋊D4), (C2×D4).70D10, C20.174(C2×D4), (C2×Q8).52D10, (C2×SD16).2D5, (C2×Dic20)⋊25C2, Dic5⋊Q84C2, (C2×Dic5).78D4, (C10×SD16).2C2, C22.264(D4×D5), C2.20(C20⋊D4), C10.29(C41D4), (C2×C40).114C22, (C2×C20).444C23, C20.17D4.8C2, (D4×C10).93C22, (Q8×C10).74C22, C2.28(SD16⋊D5), C10.48(C8.C22), (C4×Dic5).57C22, (C2×Dic10).129C22, C4.7(C2×C5⋊D4), (C2×C5⋊Q16)⋊17C2, (C2×D4.D5).9C2, (C2×C10).356(C2×D4), (C2×C4).533(C22×D5), (C2×C52C8).156C22, SmallGroup(320,794)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.31D4
C1C5C10C20C2×C20C4×Dic5Dic5⋊Q8 — C40.31D4
C5C10C2×C20 — C40.31D4
C1C22C2×C4C2×SD16

Generators and relations for C40.31D4
 G = < a,b,c | a40=b4=1, c2=a20, bab-1=a29, cac-1=a-1, cbc-1=b-1 >

Subgroups: 462 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×6], C23, C10, C10 [×2], C10, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], Q16 [×4], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×3], C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×SD16, C2×Q16 [×2], C52C8 [×2], C40 [×2], Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×C10, C8.2D4, Dic20 [×2], C2×C52C8, C4×Dic5, C10.D4 [×2], D4.D5 [×2], C5⋊Q16 [×2], C23.D5 [×2], C2×C40, C5×SD16 [×2], C2×Dic10 [×2], D4×C10, Q8×C10, C408C4, C2×Dic20, C2×D4.D5, C20.17D4, C2×C5⋊Q16, Dic5⋊Q8, C10×SD16, C40.31D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C8.C22 [×2], C5⋊D4 [×2], C22×D5, C8.2D4, D4×D5 [×2], C2×C5⋊D4, SD16⋊D5 [×2], C20⋊D4, C40.31D4

Smallest permutation representation of C40.31D4
On 160 points
Generators in S160
(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 85 132 67)(2 114 133 56)(3 103 134 45)(4 92 135 74)(5 81 136 63)(6 110 137 52)(7 99 138 41)(8 88 139 70)(9 117 140 59)(10 106 141 48)(11 95 142 77)(12 84 143 66)(13 113 144 55)(14 102 145 44)(15 91 146 73)(16 120 147 62)(17 109 148 51)(18 98 149 80)(19 87 150 69)(20 116 151 58)(21 105 152 47)(22 94 153 76)(23 83 154 65)(24 112 155 54)(25 101 156 43)(26 90 157 72)(27 119 158 61)(28 108 159 50)(29 97 160 79)(30 86 121 68)(31 115 122 57)(32 104 123 46)(33 93 124 75)(34 82 125 64)(35 111 126 53)(36 100 127 42)(37 89 128 71)(38 118 129 60)(39 107 130 49)(40 96 131 78)
(1 42 21 62)(2 41 22 61)(3 80 23 60)(4 79 24 59)(5 78 25 58)(6 77 26 57)(7 76 27 56)(8 75 28 55)(9 74 29 54)(10 73 30 53)(11 72 31 52)(12 71 32 51)(13 70 33 50)(14 69 34 49)(15 68 35 48)(16 67 36 47)(17 66 37 46)(18 65 38 45)(19 64 39 44)(20 63 40 43)(81 131 101 151)(82 130 102 150)(83 129 103 149)(84 128 104 148)(85 127 105 147)(86 126 106 146)(87 125 107 145)(88 124 108 144)(89 123 109 143)(90 122 110 142)(91 121 111 141)(92 160 112 140)(93 159 113 139)(94 158 114 138)(95 157 115 137)(96 156 116 136)(97 155 117 135)(98 154 118 134)(99 153 119 133)(100 152 120 132)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222444444455888810···1010101010202020202020202040···40
size1111822820204040224420202···28888444488884···4

44 irreducible representations

dim11111111222222224444
type+++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D5D10D10D10C5⋊D4C8.C22D4×D5D4×D5SD16⋊D5
kernelC40.31D4C408C4C2×Dic20C2×D4.D5C20.17D4C2×C5⋊Q16Dic5⋊Q8C10×SD16C52C8C40C2×Dic5C2×SD16C2×C8C2×D4C2×Q8C8C10C4C22C2
# reps11111111222222282228

Matrix representation of C40.31D4 in GL6(𝔽41)

1690000
17250000
0000360
00503636
002633238
0000288
,
1690000
17250000
00272700
00171400
002902712
0013272814
,
100000
1400000
0080233
001002131
00252100
004036233

G:=sub<GL(6,GF(41))| [16,17,0,0,0,0,9,25,0,0,0,0,0,0,0,5,26,0,0,0,0,0,33,0,0,0,36,36,23,28,0,0,0,36,8,8],[16,17,0,0,0,0,9,25,0,0,0,0,0,0,27,17,29,13,0,0,27,14,0,27,0,0,0,0,27,28,0,0,0,0,12,14],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,8,10,25,40,0,0,0,0,21,36,0,0,2,21,0,2,0,0,33,31,0,33] >;

C40.31D4 in GAP, Magma, Sage, TeX

C_{40}._{31}D_4
% in TeX

G:=Group("C40.31D4");
// GroupNames label

G:=SmallGroup(320,794);
// by ID

G=gap.SmallGroup(320,794);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,232,1094,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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