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G = C42.7D10order 320 = 26·5

7th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.7D10, C10.30C4≀C2, (D4×C10).14C4, (C2×C20).231D4, (Q8×C10).11C4, (C2×D4).1Dic5, (C2×Q8).1Dic5, C4.4D4.1D5, (C4×C20).235C22, C2.3(C20.D4), C42.D523C2, C2.6(D42Dic5), C10.15(C4.D4), C54(C42.C22), C22.39(C23.D5), (C2×C4).9(C2×Dic5), (C2×C20).342(C2×C4), (C5×C4.4D4).8C2, (C2×C4).165(C5⋊D4), (C2×C10).162(C22⋊C4), SmallGroup(320,98)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.7D10
C1C5C10C2×C10C2×C20C4×C20C42.D5 — C42.7D10
C5C2×C10C2×C20 — C42.7D10
C1C22C42C4.4D4

Generators and relations for C42.7D10
 G = < a,b,c,d | a4=b4=c10=1, d2=ab-1, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc-1 >

Subgroups: 206 in 70 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×2], C2, C4 [×4], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4, Q8, C23, C10, C10 [×2], C10, C42, C22⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, C20 [×4], C2×C10, C2×C10 [×3], C8⋊C4 [×2], C4.4D4, C52C8 [×4], C2×C20, C2×C20 [×2], C2×C20, C5×D4, C5×Q8, C22×C10, C42.C22, C2×C52C8 [×2], C4×C20, C5×C22⋊C4 [×2], D4×C10, Q8×C10, C42.D5 [×2], C5×C4.4D4, C42.7D10
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C4.D4, C4≀C2 [×2], C2×Dic5, C5⋊D4 [×2], C42.C22, C23.D5, C20.D4, D42Dic5 [×2], C42.7D10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F5A5B8A···8H10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222444444558···810···101010101020···2020202020
size111182222482220···202···288884···48888

47 irreducible representations

dim111112222222444
type++++++--+
imageC1C2C2C4C4D4D5D10Dic5Dic5C4≀C2C5⋊D4C4.D4C20.D4D42Dic5
kernelC42.7D10C42.D5C5×C4.4D4D4×C10Q8×C10C2×C20C4.4D4C42C2×D4C2×Q8C10C2×C4C10C2C2
# reps121222222288148

Matrix representation of C42.7D10 in GL6(𝔽41)

0320000
900000
009000
000900
0000400
0000040
,
010000
4000000
000100
0040000
000010
000001
,
100000
0400000
001000
0004000
00003535
0000640
,
440000
4370000
0053600
005500
0000427
00001037

G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[4,4,0,0,0,0,4,37,0,0,0,0,0,0,5,5,0,0,0,0,36,5,0,0,0,0,0,0,4,10,0,0,0,0,27,37] >;

C42.7D10 in GAP, Magma, Sage, TeX

C_4^2._7D_{10}
% in TeX

G:=Group("C4^2.7D10");
// GroupNames label

G:=SmallGroup(320,98);
// by ID

G=gap.SmallGroup(320,98);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,268,1571,570,136,12550]);
// Polycyclic

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

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