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G = C42.162D10order 320 = 26·5

162nd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.162D10, C10.1012- 1+4, C4⋊C4.213D10, C422C25D5, C42⋊D57C2, D10⋊Q844C2, (C4×Dic10)⋊15C2, (C4×C20).34C22, C22⋊C4.80D10, C4.Dic1040C2, D10.41(C4○D4), (C2×C10).252C24, (C2×C20).604C23, Dic54D4.5C2, C23.58(C22×D5), Dic5.48(C4○D4), Dic5.Q838C2, D10.12D4.4C2, C23.D1046C2, C4⋊Dic5.247C22, (C22×C10).66C23, C22.273(C23×D5), Dic5.14D446C2, C23.D5.68C22, C23.11D1022C2, (C4×Dic5).238C22, (C2×Dic5).130C23, (C22×D5).236C23, C2.65(D4.10D10), D10⋊C4.140C22, C511(C22.46C24), (C2×Dic10).263C22, C10.D4.146C22, (C22×Dic5).152C22, (D5×C4⋊C4)⋊42C2, C2.99(D5×C4○D4), C4⋊C47D541C2, (C5×C422C2)⋊7C2, C10.210(C2×C4○D4), (C2×C4×D5).271C22, (C2×C4).88(C22×D5), (C5×C4⋊C4).204C22, (C2×C5⋊D4).72C22, (C5×C22⋊C4).77C22, SmallGroup(320,1380)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.162D10
C1C5C10C2×C10C22×D5C2×C4×D5D5×C4⋊C4 — C42.162D10
C5C2×C10 — C42.162D10
C1C22C422C2

Generators and relations for C42.162D10
 G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >

Subgroups: 678 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×7], C5, C2×C4 [×6], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42, C42 [×4], C22⋊C4 [×3], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×6], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C422C2, C422C2, Dic10 [×2], C4×D5 [×6], C2×Dic5 [×7], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×6], C22×D5, C22×C10, C22.46C24, C4×Dic5 [×4], C10.D4 [×9], C4⋊Dic5 [×4], D10⋊C4 [×3], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×3], C22×Dic5, C2×C5⋊D4, C4×Dic10, C42⋊D5, C23.11D10, Dic5.14D4, C23.D10, Dic54D4, D10.12D4 [×2], Dic5.Q8 [×2], C4.Dic10, D5×C4⋊C4, C4⋊C47D5, D10⋊Q8, C5×C422C2, C42.162D10
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, C23×D5, D5×C4○D4 [×2], D4.10D10, C42.162D10

Smallest permutation representation of C42.162D10
On 160 points
Generators in S160
(1 159 67 57)(2 150 68 48)(3 141 69 59)(4 152 70 50)(5 143 71 41)(6 154 72 52)(7 145 73 43)(8 156 74 54)(9 147 75 45)(10 158 76 56)(11 149 77 47)(12 160 78 58)(13 151 79 49)(14 142 80 60)(15 153 61 51)(16 144 62 42)(17 155 63 53)(18 146 64 44)(19 157 65 55)(20 148 66 46)(21 101 140 86)(22 112 121 97)(23 103 122 88)(24 114 123 99)(25 105 124 90)(26 116 125 81)(27 107 126 92)(28 118 127 83)(29 109 128 94)(30 120 129 85)(31 111 130 96)(32 102 131 87)(33 113 132 98)(34 104 133 89)(35 115 134 100)(36 106 135 91)(37 117 136 82)(38 108 137 93)(39 119 138 84)(40 110 139 95)
(1 36 11 26)(2 126 12 136)(3 38 13 28)(4 128 14 138)(5 40 15 30)(6 130 16 140)(7 22 17 32)(8 132 18 122)(9 24 19 34)(10 134 20 124)(21 72 31 62)(23 74 33 64)(25 76 35 66)(27 78 37 68)(29 80 39 70)(41 95 51 85)(42 101 52 111)(43 97 53 87)(44 103 54 113)(45 99 55 89)(46 105 56 115)(47 81 57 91)(48 107 58 117)(49 83 59 93)(50 109 60 119)(61 129 71 139)(63 131 73 121)(65 133 75 123)(67 135 77 125)(69 137 79 127)(82 150 92 160)(84 152 94 142)(86 154 96 144)(88 156 98 146)(90 158 100 148)(102 145 112 155)(104 147 114 157)(106 149 116 159)(108 151 118 141)(110 153 120 143)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 129 31 139)(22 138 32 128)(23 127 33 137)(24 136 34 126)(25 125 35 135)(26 134 36 124)(27 123 37 133)(28 132 38 122)(29 121 39 131)(30 130 40 140)(41 42 51 52)(43 60 53 50)(44 49 54 59)(45 58 55 48)(46 47 56 57)(61 62 71 72)(63 80 73 70)(64 69 74 79)(65 78 75 68)(66 67 76 77)(81 105 91 115)(82 114 92 104)(83 103 93 113)(84 112 94 102)(85 101 95 111)(86 110 96 120)(87 119 97 109)(88 108 98 118)(89 117 99 107)(90 106 100 116)(141 146 151 156)(142 155 152 145)(143 144 153 154)(147 160 157 150)(148 149 158 159)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I···4N4O4P4Q4R5A5B10A···10F10G10H20A···20L20M···20R
order1222222444444444···444445510···10101020···2020···20
size1111410102222444410···1020202020222···2884···48···8

53 irreducible representations

dim11111111111111222222444
type++++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D102- 1+4D5×C4○D4D4.10D10
kernelC42.162D10C4×Dic10C42⋊D5C23.11D10Dic5.14D4C23.D10Dic54D4D10.12D4Dic5.Q8C4.Dic10D5×C4⋊C4C4⋊C47D5D10⋊Q8C5×C422C2C422C2Dic5D10C42C22⋊C4C4⋊C4C10C2C2
# reps11111112211111244266184

Matrix representation of C42.162D10 in GL6(𝔽41)

4000000
0400000
0032000
0003200
00003223
000099
,
100000
010000
000100
001000
00004039
000011
,
660000
3510000
0040000
000100
00001222
00004029
,
660000
1350000
0040000
0004000
00001222
00004029

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,9,0,0,0,0,23,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,12,40,0,0,0,0,22,29],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,40,0,0,0,0,22,29] >;

C42.162D10 in GAP, Magma, Sage, TeX

C_4^2._{162}D_{10}
% in TeX

G:=Group("C4^2.162D10");
// GroupNames label

G:=SmallGroup(320,1380);
// by ID

G=gap.SmallGroup(320,1380);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,570,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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