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G = C42.90D10order 320 = 26·5

90th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.90D10, C10.922+ 1+4, C10.482- 1+4, (C2×C20)⋊5Q8, C20⋊Q811C2, C202Q86C2, (C2×C4)⋊4Dic10, C20.78(C2×Q8), C4⋊C4.268D10, (C4×C20).7C22, C20.6Q84C2, (C2×C10).63C24, C22⋊C4.91D10, C4.Dic1011C2, C4.34(C2×Dic10), C2.6(D48D10), C10.11(C22×Q8), (C2×C20).142C23, C42⋊C2.13D5, (C22×C4).187D10, C4⋊Dic5.32C22, C22.7(C2×Dic10), C22.96(C23×D5), (C2×Dic5).22C23, (C4×Dic5).76C22, C2.13(C22×Dic10), C10.D4.2C22, C23.151(C22×D5), C2.7(D4.10D10), C23.D5.92C22, (C22×C20).223C22, (C22×C10).133C23, Dic5.14D4.1C2, C52(C23.41C23), (C2×Dic10).25C22, C23.21D10.23C2, (C22×Dic5).85C22, (C2×C10).13(C2×Q8), (C2×C4⋊Dic5).45C2, (C5×C4⋊C4).304C22, (C2×C4).148(C22×D5), (C5×C42⋊C2).14C2, (C5×C22⋊C4).99C22, SmallGroup(320,1191)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.90D10
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C4⋊Dic5 — C42.90D10
C5C2×C10 — C42.90D10
C1C22C42⋊C2

Generators and relations for C42.90D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 638 in 206 conjugacy classes, 111 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×18], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.41C23, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×2], C4⋊Dic5 [×8], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C202Q8 [×2], C20.6Q8 [×2], Dic5.14D4 [×4], C20⋊Q8 [×2], C4.Dic10 [×2], C2×C4⋊Dic5, C23.21D10, C5×C42⋊C2, C42.90D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4, 2- 1+4, Dic10 [×4], C22×D5 [×7], C23.41C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D48D10, D4.10D10, C42.90D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4P5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order122222444444444···45510···101010101020···2020···20
size1111222222444420···20222···244442···24···4

62 irreducible representations

dim11111111122222224444
type+++++++++-+++++-+-+-
imageC1C2C2C2C2C2C2C2C2Q8D5D10D10D10D10Dic102+ 1+42- 1+4D48D10D4.10D10
kernelC42.90D10C202Q8C20.6Q8Dic5.14D4C20⋊Q8C4.Dic10C2×C4⋊Dic5C23.21D10C5×C42⋊C2C2×C20C42⋊C2C42C22⋊C4C4⋊C4C22×C4C2×C4C10C10C2C2
# reps122422111424442161144

Matrix representation of C42.90D10 in GL6(𝔽41)

010000
4000000
0000400
0000040
0040000
0004000
,
100000
010000
0021300
00283900
0000213
00002839
,
4000000
0400000
00353500
0064000
000066
0000351
,
090000
900000
00371400
0031400
00003714
0000314

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,37,31,0,0,0,0,14,4,0,0,0,0,0,0,37,31,0,0,0,0,14,4] >;

C42.90D10 in GAP, Magma, Sage, TeX

C_4^2._{90}D_{10}
% in TeX

G:=Group("C4^2.90D10");
// GroupNames label

G:=SmallGroup(320,1191);
// by ID

G=gap.SmallGroup(320,1191);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,184,675,570,80,12550]);
// Polycyclic

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

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