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G = C42.92D10order 320 = 26·5

92nd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.92D10, C10.502- 1+4, C202Q87C2, C4.72(C2×D20), (C2×C4).56D20, C4⋊C4.269D10, (C2×C20).202D4, C4.D204C2, C20.288(C2×D4), (C4×C20).8C22, D102Q811C2, C42⋊C210D5, (C2×C10).70C24, C22⋊C4.94D10, C22.21(C2×D20), C10.14(C22×D4), C2.16(C22×D20), (C2×C20).145C23, C22.D204C2, (C22×C4).191D10, C4⋊Dic5.33C22, C22.99(C23×D5), (C22×Dic10)⋊15C2, D10⋊C4.2C22, (C2×D20).214C22, (C2×Dic5).24C23, (C22×D5).20C23, C23.158(C22×D5), C2.8(D4.10D10), (C22×C10).140C23, (C22×C20).230C22, C51(C23.38C23), (C2×Dic10).293C22, (C22×Dic5).87C22, (C2×C10).51(C2×D4), (C2×C4×D5).68C22, (C2×C4○D20).19C2, (C5×C42⋊C2)⋊12C2, (C5×C4⋊C4).307C22, (C2×C4).576(C22×D5), (C2×C5⋊D4).109C22, (C5×C22⋊C4).102C22, SmallGroup(320,1198)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.92D10
C1C5C10C2×C10C22×D5C2×C4×D5C2×C4○D20 — C42.92D10
C5C2×C10 — C42.92D10
C1C22C42⋊C2

Generators and relations for C42.92D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, dbd-1=b-1, dcd-1=c9 >

Subgroups: 974 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8 [×9], C4○D4 [×4], Dic5 [×6], C20 [×4], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×10], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×C10, C23.38C23, C4⋊Dic5 [×8], D10⋊C4 [×8], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×4], C2×Dic10 [×4], C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20, C202Q8 [×2], C4.D20 [×2], C22.D20 [×4], D102Q8 [×4], C5×C42⋊C2, C22×Dic10, C2×C4○D20, C42.92D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- 1+4 [×2], D20 [×4], C22×D5 [×7], C23.38C23, C2×D20 [×6], C23×D5, C22×D20, D4.10D10 [×2], C42.92D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I···4N5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order12222222444444444···45510···101010101020···2020···20
size11112220202222444420···20222···244442···24···4

62 irreducible representations

dim11111111222222244
type+++++++++++++++--
imageC1C2C2C2C2C2C2C2D4D5D10D10D10D10D202- 1+4D4.10D10
kernelC42.92D10C202Q8C4.D20C22.D20D102Q8C5×C42⋊C2C22×Dic10C2×C4○D20C2×C20C42⋊C2C42C22⋊C4C4⋊C4C22×C4C2×C4C10C2
# reps122441114244421628

Matrix representation of C42.92D10 in GL6(𝔽41)

100000
010000
0030900
00321100
0000309
00003211
,
1390000
1400000
00309390
003211039
00001132
0000930
,
100000
010000
001225351
001623401
0012142916
0027282518
,
100000
1400000
0023162528
0013183916
00001225
00002729

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,32,0,0,0,0,9,11,0,0,0,0,0,0,30,32,0,0,0,0,9,11],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,30,32,0,0,0,0,9,11,0,0,0,0,39,0,11,9,0,0,0,39,32,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,16,12,27,0,0,25,23,14,28,0,0,35,40,29,25,0,0,1,1,16,18],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,23,13,0,0,0,0,16,18,0,0,0,0,25,39,12,27,0,0,28,16,25,29] >;

C42.92D10 in GAP, Magma, Sage, TeX

C_4^2._{92}D_{10}
% in TeX

G:=Group("C4^2.92D10");
// GroupNames label

G:=SmallGroup(320,1198);
// by ID

G=gap.SmallGroup(320,1198);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,297,192,12550]);
// Polycyclic

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

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