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G = C42.185D10order 320 = 26·5

5th non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.185D10, C8⋊C49D5, (C2×C8).158D10, C10.47(C8○D4), C20.8Q837C2, D101C8.16C2, D10⋊C4.18C4, C4.131(C4○D20), C20.247(C4○D4), (C4×C20).230C22, (C2×C20).815C23, (C2×C40).312C22, C10.D4.18C4, C42⋊D5.12C2, C2.8(D20.2C4), C2.13(C42⋊D5), C54(C42.7C22), C10.29(C42⋊C2), (C4×Dic5).201C22, (C4×C52C8)⋊22C2, (C5×C8⋊C4)⋊18C2, (C2×C4).62(C4×D5), C22.100(C2×C4×D5), (C2×C20).322(C2×C4), (C2×C4×D5).229C22, (C2×Dic5).17(C2×C4), (C22×D5).16(C2×C4), (C2×C4).757(C22×D5), (C2×C10).171(C22×C4), (C2×C52C8).305C22, SmallGroup(320,336)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.185D10
C1C5C10C20C2×C20C2×C4×D5C42⋊D5 — C42.185D10
C5C2×C10 — C42.185D10
C1C2×C4C8⋊C4

Generators and relations for C42.185D10
 G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 302 in 96 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×5], C23, D5, C10, C10 [×2], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C42⋊C2, C52C8 [×2], C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C42.7C22, C2×C52C8 [×2], C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C4×C52C8, C20.8Q8 [×2], D101C8 [×2], C5×C8⋊C4, C42⋊D5, C42.185D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C8○D4 [×2], C4×D5 [×2], C22×D5, C42.7C22, C2×C4×D5, C4○D20 [×2], C42⋊D5, D20.2C4 [×2], C42.185D10

Smallest permutation representation of C42.185D10
On 160 points
Generators in S160
(1 63 153 92)(2 44 154 113)(3 65 155 94)(4 46 156 115)(5 67 157 96)(6 48 158 117)(7 69 159 98)(8 50 160 119)(9 71 121 100)(10 52 122 81)(11 73 123 102)(12 54 124 83)(13 75 125 104)(14 56 126 85)(15 77 127 106)(16 58 128 87)(17 79 129 108)(18 60 130 89)(19 41 131 110)(20 62 132 91)(21 43 133 112)(22 64 134 93)(23 45 135 114)(24 66 136 95)(25 47 137 116)(26 68 138 97)(27 49 139 118)(28 70 140 99)(29 51 141 120)(30 72 142 101)(31 53 143 82)(32 74 144 103)(33 55 145 84)(34 76 146 105)(35 57 147 86)(36 78 148 107)(37 59 149 88)(38 80 150 109)(39 61 151 90)(40 42 152 111)
(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 10 123 132 21 30 143 152)(2 131 124 29 22 151 144 9)(3 28 125 150 23 8 145 130)(4 149 126 7 24 129 146 27)(5 6 127 128 25 26 147 148)(11 20 133 142 31 40 153 122)(12 141 134 39 32 121 154 19)(13 38 135 160 33 18 155 140)(14 159 136 17 34 139 156 37)(15 16 137 138 35 36 157 158)(41 54 120 93 61 74 100 113)(42 92 81 73 62 112 101 53)(43 72 82 111 63 52 102 91)(44 110 83 51 64 90 103 71)(45 50 84 89 65 70 104 109)(46 88 85 69 66 108 105 49)(47 68 86 107 67 48 106 87)(55 60 94 99 75 80 114 119)(56 98 95 79 76 118 115 59)(57 78 96 117 77 58 116 97)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D8E···8L10A···10F20A···20H20I···20P40A···40P
order12222444444444445588888···810···1020···2020···2040···40
size1111201111222220202022444410···102···22···24···44···4

68 irreducible representations

dim1111111122222224
type+++++++++
imageC1C2C2C2C2C2C4C4D5C4○D4D10D10C8○D4C4×D5C4○D20D20.2C4
kernelC42.185D10C4×C52C8C20.8Q8D101C8C5×C8⋊C4C42⋊D5C10.D4D10⋊C4C8⋊C4C20C42C2×C8C10C2×C4C4C2
# reps11221144242488168

Matrix representation of C42.185D10 in GL6(𝔽41)

100000
010000
00403100
000100
0000935
0000032
,
100000
010000
0032000
0003200
000090
000009
,
770000
34400000
0014000
00302700
00001418
0000127
,
770000
40340000
0014000
0001400
00002723
0000014

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,31,1,0,0,0,0,0,0,9,0,0,0,0,0,35,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,14,30,0,0,0,0,0,27,0,0,0,0,0,0,14,1,0,0,0,0,18,27],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,27,0,0,0,0,0,23,14] >;

C42.185D10 in GAP, Magma, Sage, TeX

C_4^2._{185}D_{10}
% in TeX

G:=Group("C4^2.185D10");
// GroupNames label

G:=SmallGroup(320,336);
// by ID

G=gap.SmallGroup(320,336);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,422,387,58,136,12550]);
// 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=a*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations

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