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G = C42.62D10order 320 = 26·5

62nd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.62D10, (C2×C20).81D4, C202Q817C2, (C2×D4).45D10, (C2×Q8).35D10, C4.4D4.5D5, C20.66(C4○D4), Q8⋊Dic520C2, C42.D58C2, C4.20(D42D5), (C2×C20).373C23, (C4×C20).104C22, D4⋊Dic5.12C2, (D4×C10).61C22, (Q8×C10).53C22, C2.17(D4⋊D10), C10.118(C8⋊C22), C10.41(C4.4D4), C2.8(C20.17D4), C4⋊Dic5.150C22, C2.18(D4.9D10), C10.119(C8.C22), C54(C42.28C22), (C2×C10).504(C2×D4), (C2×C4).60(C5⋊D4), (C5×C4.4D4).3C2, (C2×C4).473(C22×D5), C22.179(C2×C5⋊D4), (C2×C52C8).120C22, SmallGroup(320,682)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.62D10
C1C5C10C2×C10C2×C20C2×C52C8C42.D5 — C42.62D10
C5C10C2×C20 — C42.62D10
C1C22C42C4.4D4

Generators and relations for C42.62D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 350 in 100 conjugacy classes, 39 normal (27 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, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×D4, C2×Q8, C2×Q8, Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×3], C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×C10, C42.28C22, C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C5×C22⋊C4 [×2], C2×Dic10, D4×C10, Q8×C10, C42.D5, D4⋊Dic5 [×2], Q8⋊Dic5 [×2], C202Q8, C5×C4.4D4, C42.62D10
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, D42D5 [×2], C2×C5⋊D4, C20.17D4, D4⋊D10, D4.9D10, C42.62D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222444444455888810···101010101020···2020202020
size1111822448404022202020202···288884···48888

44 irreducible representations

dim111111222222244444
type++++++++++++--+-
imageC1C2C2C2C2C2D4D5C4○D4D10D10D10C5⋊D4C8⋊C22C8.C22D42D5D4⋊D10D4.9D10
kernelC42.62D10C42.D5D4⋊Dic5Q8⋊Dic5C202Q8C5×C4.4D4C2×C20C4.4D4C20C42C2×D4C2×Q8C2×C4C10C10C4C2C2
# reps112211224222811444

Matrix representation of C42.62D10 in GL8(𝔽41)

00100000
00010000
400000000
040000000
000030900
0000321100
0000250119
00000163230
,
10000000
01000000
00100000
00010000
000026271322
00000402813
00002421114
00002424015
,
77000000
3440000000
0034340000
00710000
000040700
000034700
000036333434
0000403671
,
003200000
002290000
320000000
229000000
00000303110
00003551039
00004036300
0000212106

G:=sub<GL(8,GF(41))| [0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30,32,25,0,0,0,0,0,9,11,0,16,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,26,0,24,24,0,0,0,0,27,40,21,24,0,0,0,0,13,28,1,0,0,0,0,0,22,13,14,15],[7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,40,34,36,40,0,0,0,0,7,7,33,36,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1],[0,0,32,22,0,0,0,0,0,0,0,9,0,0,0,0,32,22,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,35,40,21,0,0,0,0,30,5,36,21,0,0,0,0,31,10,30,0,0,0,0,0,10,39,0,6] >;

C42.62D10 in GAP, Magma, Sage, TeX

C_4^2._{62}D_{10}
% in TeX

G:=Group("C4^2.62D10");
// GroupNames label

G:=SmallGroup(320,682);
// by ID

G=gap.SmallGroup(320,682);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,590,471,438,102,12550]);
// 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*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations

׿
×
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