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G = C42.105D10order 320 = 26·5

105th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.105D10, C10.562- 1+4, (C4×D4).12D5, C4⋊C4.280D10, (D4×C20).13C2, (C4×Dic10)⋊27C2, (C2×D4).209D10, C20.48D48C2, (C2×C10).85C24, C20.6Q815C2, C20.291(C4○D4), (C4×C20).147C22, (C2×C20).585C23, C22⋊C4.130D10, Dic5.Q87C2, (C22×C4).204D10, C23.D106C2, C4.115(D42D5), C23.D5.9C22, (D4×C10).303C22, C22.11(C4○D20), C23.21D106C2, C4⋊Dic5.296C22, (C22×C20).79C22, (C2×Dic5).35C23, C22.113(C23×D5), C23.165(C22×D5), C23.11D1027C2, (C22×C10).155C23, C54(C22.46C24), (C4×Dic5).222C22, C23.18D10.5C2, C2.14(D4.10D10), (C2×Dic10).244C22, C10.D4.153C22, (C22×Dic5).93C22, (C2×C4⋊Dic5)⋊23C2, C2.41(C2×C4○D20), C10.37(C2×C4○D4), C2.19(C2×D42D5), (C2×C10).15(C4○D4), (C5×C4⋊C4).321C22, (C2×C4).155(C22×D5), (C5×C22⋊C4).142C22, SmallGroup(320,1213)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.105D10
C1C5C10C2×C10C2×Dic5C22×Dic5C23.11D10 — C42.105D10
C5C2×C10 — C42.105D10
C1C22C4×D4

Generators and relations for C42.105D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 598 in 214 conjugacy classes, 99 normal (51 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C5, C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×2], C23 [×2], C10 [×3], C10 [×3], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×Q8, Dic5 [×8], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C422C2 [×2], Dic10 [×2], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C22.46C24, C4×Dic5 [×4], C10.D4 [×10], C4⋊Dic5 [×3], C4⋊Dic5 [×2], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C20.6Q8, C23.11D10 [×2], C23.D10 [×2], Dic5.Q8 [×2], C20.48D4 [×2], C2×C4⋊Dic5, C23.21D10, C23.18D10 [×2], D4×C20, C42.105D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.46C24, C4○D20 [×2], D42D5 [×2], C23×D5, C2×C4○D20, C2×D42D5, D4.10D10, C42.105D10

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

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G4H4I4J4K4L4M···4R5A5B10A···10F10G···10N20A···20H20I···20X
order12222224···44444444···45510···1010···1020···2020···20
size11112242···2441010101020···20222···24···42···24···4

65 irreducible representations

dim11111111111222222222444
type+++++++++++++++++---
imageC1C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D10D10C4○D202- 1+4D42D5D4.10D10
kernelC42.105D10C4×Dic10C20.6Q8C23.11D10C23.D10Dic5.Q8C20.48D4C2×C4⋊Dic5C23.21D10C23.18D10D4×C20C4×D4C20C2×C10C42C22⋊C4C4⋊C4C22×C4C2×D4C22C10C4C2
# reps111222211212442424216144

Matrix representation of C42.105D10 in GL4(𝔽41) generated by

40000
0100
00412
00237
,
32000
0900
00400
00040
,
25000
02300
003728
00394
,
02300
25000
0056
002336
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,4,2,0,0,12,37],[32,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[25,0,0,0,0,23,0,0,0,0,37,39,0,0,28,4],[0,25,0,0,23,0,0,0,0,0,5,23,0,0,6,36] >;

C42.105D10 in GAP, Magma, Sage, TeX

C_4^2._{105}D_{10}
% in TeX

G:=Group("C4^2.105D10");
// GroupNames label

G:=SmallGroup(320,1213);
// by ID

G=gap.SmallGroup(320,1213);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,675,570,12550]);
// 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=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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