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G = C42.182D10order 320 = 26·5

2nd non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.182D10, D10.5M4(2), Dic5.8M4(2), C8⋊C47D5, (C2×C8).156D10, C2.9(D5×M4(2)), C55(C42.6C4), C20.8Q836C2, (C4×Dic5).18C4, (D5×C42).14C2, D101C8.15C2, C20.245(C4○D4), C4.129(C4○D20), (C2×C40).311C22, (C4×C20).227C22, (C2×C20).811C23, C42.D518C2, C10.50(C2×M4(2)), C2.12(C42⋊D5), C10.28(C42⋊C2), (C4×Dic5).298C22, (C2×C4×D5).19C4, (C5×C8⋊C4)⋊17C2, C22.98(C2×C4×D5), (C2×C4).127(C4×D5), (C2×C20).319(C2×C4), (C2×C4×D5).339C22, (C22×D5).96(C2×C4), (C2×C4).753(C22×D5), (C2×C10).167(C22×C4), (C2×C52C8).193C22, (C2×Dic5).136(C2×C4), SmallGroup(320,332)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.182D10
C1C5C10C20C2×C20C2×C4×D5D5×C42 — C42.182D10
C5C2×C10 — C42.182D10
C1C2×C4C8⋊C4

Generators and relations for C42.182D10
 G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 350 in 110 conjugacy classes, 51 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×9], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×2], C2×C8 [×2], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8⋊C4, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C52C8 [×2], C40 [×2], C4×D5 [×6], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.6C4, C2×C52C8 [×2], C4×Dic5 [×3], C4×C20, C2×C40 [×2], C2×C4×D5 [×3], C42.D5, C20.8Q8 [×2], D101C8 [×2], C5×C8⋊C4, D5×C42, C42.182D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, M4(2) [×4], C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C2×M4(2) [×2], C4×D5 [×2], C22×D5, C42.6C4, C2×C4×D5, C4○D20 [×2], C42⋊D5, D5×M4(2) [×2], C42.182D10

Smallest permutation representation of C42.182D10
On 160 points
Generators in S160
(1 81 122 43)(2 102 123 64)(3 83 124 45)(4 104 125 66)(5 85 126 47)(6 106 127 68)(7 87 128 49)(8 108 129 70)(9 89 130 51)(10 110 131 72)(11 91 132 53)(12 112 133 74)(13 93 134 55)(14 114 135 76)(15 95 136 57)(16 116 137 78)(17 97 138 59)(18 118 139 80)(19 99 140 61)(20 120 141 42)(21 101 142 63)(22 82 143 44)(23 103 144 65)(24 84 145 46)(25 105 146 67)(26 86 147 48)(27 107 148 69)(28 88 149 50)(29 109 150 71)(30 90 151 52)(31 111 152 73)(32 92 153 54)(33 113 154 75)(34 94 155 56)(35 115 156 77)(36 96 157 58)(37 117 158 79)(38 98 159 60)(39 119 160 41)(40 100 121 62)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 10 132 141 21 30 152 121)(2 140 133 29 22 160 153 9)(3 28 134 159 23 8 154 139)(4 158 135 7 24 138 155 27)(5 6 136 137 25 26 156 157)(11 20 142 151 31 40 122 131)(12 150 143 39 32 130 123 19)(13 38 144 129 33 18 124 149)(14 128 145 17 34 148 125 37)(15 16 146 147 35 36 126 127)(41 74 89 82 61 54 109 102)(42 81 90 53 62 101 110 73)(43 52 91 100 63 72 111 120)(44 99 92 71 64 119 112 51)(45 70 93 118 65 50 113 98)(46 117 94 49 66 97 114 69)(47 48 95 96 67 68 115 116)(55 80 103 88 75 60 83 108)(56 87 104 59 76 107 84 79)(57 58 105 106 77 78 85 86)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H20I···20P40A···40P
order122222444444444···4558888888810···1020···2020···2040···40
size111110101111222210···10224444202020202···22···24···44···4

68 irreducible representations

dim11111111222222224
type+++++++++
imageC1C2C2C2C2C2C4C4D5M4(2)C4○D4M4(2)D10D10C4×D5C4○D20D5×M4(2)
kernelC42.182D10C42.D5C20.8Q8D101C8C5×C8⋊C4D5×C42C4×Dic5C2×C4×D5C8⋊C4Dic5C20D10C42C2×C8C2×C4C4C2
# reps112211442444248168

Matrix representation of C42.182D10 in GL4(𝔽41) generated by

323800
0900
0090
0009
,
9000
0900
00400
00040
,
13100
42800
001427
001430
,
131700
42800
002714
003014
G:=sub<GL(4,GF(41))| [32,0,0,0,38,9,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[13,4,0,0,1,28,0,0,0,0,14,14,0,0,27,30],[13,4,0,0,17,28,0,0,0,0,27,30,0,0,14,14] >;

C42.182D10 in GAP, Magma, Sage, TeX

C_4^2._{182}D_{10}
% in TeX

G:=Group("C4^2.182D10");
// GroupNames label

G:=SmallGroup(320,332);
// by ID

G=gap.SmallGroup(320,332);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,422,387,58,136,12550]);
// Polycyclic

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

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