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G = C42.61D10order 320 = 26·5

61st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.61D10, Dic10.23D4, C4.49(D4×D5), C203C828C2, C20.23(C2×D4), (C2×D4).44D10, (C2×C20).269D4, (C2×Q8).34D10, C55(Q8.D4), C4.4D4.4D5, (C4×Dic10)⋊21C2, C20.65(C4○D4), C4.1(D42D5), Q8⋊Dic519C2, C10.103(C4○D8), C2.10(C202D4), (C2×C20).372C23, (C4×C20).103C22, D4⋊Dic5.11C2, (D4×C10).60C22, (Q8×C10).52C22, C10.101(C4⋊D4), C4⋊Dic5.340C22, C2.22(D4.8D10), C2.17(D4.9D10), C10.118(C8.C22), (C2×Dic10).280C22, (C2×C5⋊Q16)⋊12C2, (C2×D4.D5).6C2, (C2×C10).503(C2×D4), (C2×C4).59(C5⋊D4), (C5×C4.4D4).2C2, (C2×C4).472(C22×D5), C22.178(C2×C5⋊D4), (C2×C52C8).119C22, SmallGroup(320,681)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.61D10
C1C5C10C20C2×C20C4⋊Dic5C4×Dic10 — C42.61D10
C5C10C2×C20 — C42.61D10
C1C22C42C4.4D4

Generators and relations for C42.61D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 374 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×5], C23, C10 [×3], C10, C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C2×Q8, Dic5 [×3], C20 [×2], C20 [×3], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×C10, Q8.D4, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, D4.D5 [×2], C5⋊Q16 [×2], C4×C20, C5×C22⋊C4 [×2], C2×Dic10, D4×C10, Q8×C10, C203C8, D4⋊Dic5, Q8⋊Dic5, C4×Dic10, C2×D4.D5, C2×C5⋊Q16, C5×C4.4D4, C42.61D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8.C22, C5⋊D4 [×2], C22×D5, Q8.D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, D4.8D10, D4.9D10, C42.61D10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222444444444455888810···101010101020···2020202020
size111182222482020202022202020202···288884···48888

47 irreducible representations

dim1111111122222222244444
type++++++++++++++-+--
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C5⋊D4C8.C22D4×D5D42D5D4.8D10D4.9D10
kernelC42.61D10C203C8D4⋊Dic5Q8⋊Dic5C4×Dic10C2×D4.D5C2×C5⋊Q16C5×C4.4D4Dic10C2×C20C4.4D4C20C42C2×D4C2×Q8C10C2×C4C10C4C4C2C2
# reps1111111122222224812244

Matrix representation of C42.61D10 in GL6(𝔽41)

4000000
0400000
0011800
0094000
000090
000009
,
4000000
0400000
001000
000100
0000120
0000440
,
1600000
13230000
001000
0094000
0000120
0000040
,
10180000
24310000
009000
00403200
000006
00003424

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,9,0,0,0,0,18,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,20,40],[16,13,0,0,0,0,0,23,0,0,0,0,0,0,1,9,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,20,40],[10,24,0,0,0,0,18,31,0,0,0,0,0,0,9,40,0,0,0,0,0,32,0,0,0,0,0,0,0,34,0,0,0,0,6,24] >;

C42.61D10 in GAP, Magma, Sage, TeX

C_4^2._{61}D_{10}
% in TeX

G:=Group("C4^2.61D10");
// GroupNames label

G:=SmallGroup(320,681);
// by ID

G=gap.SmallGroup(320,681);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,254,219,1123,297,136,12550]);
// Polycyclic

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

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