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G = C42.108D10order 320 = 26·5

108th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.108D10, C10.162+ 1+4, C10.582- 1+4, (C4×D4)⋊8D5, D410(C4×D5), D42D55C4, (D4×C20)⋊10C2, (D4×Dic5)⋊9C2, C4⋊C4.315D10, Dic1022(C2×C4), (C4×Dic10)⋊30C2, (C2×D4).246D10, C42⋊D512C2, C20.68(C22×C4), C10.44(C23×C4), (C2×C10).90C24, (C22×C4).46D10, Dic54D446C2, Dic53Q815C2, C2.4(D46D10), (C2×C20).491C23, (C4×C20).150C22, C22⋊C4.131D10, D10.18(C22×C4), C22.33(C23×D5), (D4×C10).254C22, C4⋊Dic5.362C22, Dic5.40(C22×C4), C23.169(C22×D5), C2.3(D4.10D10), C23.11D1028C2, (C22×C10).160C23, (C22×C20).363C22, C54(C23.33C23), (C4×Dic5).223C22, (C2×Dic5).212C23, (C22×D5).179C23, C23.D5.104C22, D10⋊C4.121C22, (C2×Dic10).295C22, C10.D4.134C22, (C22×Dic5).95C22, C4.33(C2×C4×D5), (D5×C4⋊C4)⋊14C2, (C4×D5)⋊4(C2×C4), C5⋊D48(C2×C4), (C5×D4)⋊23(C2×C4), (C4×C5⋊D4)⋊41C2, C22.3(C2×C4×D5), C2.25(D5×C22×C4), (C2×C4×D5).72C22, (C2×Dic5)⋊13(C2×C4), (C2×D42D5).9C2, (C2×C10.D4)⋊38C2, (C5×C4⋊C4).324C22, (C2×C10).10(C22×C4), (C2×C4).283(C22×D5), (C2×C5⋊D4).119C22, (C5×C22⋊C4).143C22, SmallGroup(320,1218)

Series: Derived Chief Lower central Upper central

C1C10 — C42.108D10
C1C5C10C2×C10C22×D5C2×C4×D5C2×D42D5 — C42.108D10
C5C10 — C42.108D10
C1C22C4×D4

Generators and relations for C42.108D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 862 in 294 conjugacy classes, 151 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×14], C22, C22 [×4], C22 [×8], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×25], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×4], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4, C4×D4 [×5], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×12], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×C10 [×2], C23.33C23, C4×Dic5, C4×Dic5 [×4], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], D42D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C42⋊D5, C23.11D10 [×2], Dic54D4 [×2], Dic53Q8, D5×C4⋊C4, C2×C10.D4 [×2], C4×C5⋊D4 [×2], D4×Dic5, D4×C20, C2×D42D5, C42.108D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ 1+4, 2- 1+4, C4×D5 [×4], C22×D5 [×7], C23.33C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D46D10, D4.10D10, C42.108D10

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

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4J4K···4X5A5B10A···10F10G···10N20A···20H20I···20X
order12222222224···44···45510···1010···1020···2020···20
size1111222210102···210···10222···24···42···24···4

74 irreducible representations

dim111111111111122222224444
type+++++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D5D10D10D10D10D10C4×D52+ 1+42- 1+4D46D10D4.10D10
kernelC42.108D10C4×Dic10C42⋊D5C23.11D10Dic54D4Dic53Q8D5×C4⋊C4C2×C10.D4C4×C5⋊D4D4×Dic5D4×C20C2×D42D5D42D5C4×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D4C10C10C2C2
# reps11122112211116224242161144

Matrix representation of C42.108D10 in GL6(𝔽41)

4000000
0400000
00400220
00040022
0026010
0002601
,
3200000
0320000
0017100
00402400
0000171
00004024
,
0340000
6350000
00525267
0016163434
007353616
00662525
,
6340000
5350000
00255726
0016163434
003571636
00662525

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,26,0,0,0,0,40,0,26,0,0,22,0,1,0,0,0,0,22,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,17,40,0,0,0,0,1,24],[0,6,0,0,0,0,34,35,0,0,0,0,0,0,5,16,7,6,0,0,25,16,35,6,0,0,26,34,36,25,0,0,7,34,16,25],[6,5,0,0,0,0,34,35,0,0,0,0,0,0,25,16,35,6,0,0,5,16,7,6,0,0,7,34,16,25,0,0,26,34,36,25] >;

C42.108D10 in GAP, Magma, Sage, TeX

C_4^2._{108}D_{10}
% in TeX

G:=Group("C4^2.108D10");
// GroupNames label

G:=SmallGroup(320,1218);
// by ID

G=gap.SmallGroup(320,1218);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,570,80,12550]);
// Polycyclic

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

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