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G = C10.90(C4×D4)  order 320 = 26·5

16th non-split extension by C10 of C4×D4 acting via C4×D4/C4⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.90(C4×D4), (C2×C20).252D4, D10⋊C410C4, C22.113(D4×D5), (C22×C4).43D10, C10.91(C4⋊D4), (C2×Dic5).234D4, C2.20(D208C4), C2.6(Dic5⋊D4), C10.52(C4.4D4), C2.3(C20.23D4), C22.59(C4○D20), (C22×C20).30C22, C58(C24.C22), (C23×D5).19C22, C23.298(C22×D5), C10.10C4242C2, C10.57(C42⋊C2), C10.28(C422C2), C2.7(D10.13D4), C22.60(D42D5), (C22×C10).354C23, C22.30(Q82D5), C10.52(C22.D4), (C22×Dic5).216C22, (C2×C4⋊C4)⋊8D5, (C10×C4⋊C4)⋊25C2, (C2×C4×Dic5)⋊25C2, (C2×C4).42(C4×D5), C2.14(C4×C5⋊D4), C22.139(C2×C4×D5), (C2×C20).359(C2×C4), C2.7(C4⋊C4⋊D5), (C2×C10).335(C2×D4), C22.69(C2×C5⋊D4), C2.14(C4⋊C47D5), (C2×C4).100(C5⋊D4), (C22×D5).27(C2×C4), (C2×D10⋊C4).15C2, (C2×C10).157(C4○D4), (C2×C10).223(C22×C4), (C2×Dic5).109(C2×C4), SmallGroup(320,617)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.90(C4×D4)
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — C10.90(C4×D4)
C5C2×C10 — C10.90(C4×D4)
C1C23C2×C4⋊C4

Generators and relations for C10.90(C4×D4)
 G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=dad-1=a-1, ac=ca, cbc-1=a5b, bd=db, dcd-1=a5c-1 >

Subgroups: 750 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×10], C22 [×7], C22 [×10], C5, C2×C4 [×4], C2×C4 [×18], C23, C23 [×8], D5 [×2], C10 [×7], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×3], C24, Dic5 [×5], C20 [×5], D10 [×10], C2×C10 [×7], C2.C42 [×2], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×4], C2×C20 [×7], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.C22, C4×Dic5 [×2], D10⋊C4 [×4], D10⋊C4 [×4], C5×C4⋊C4 [×2], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42 [×2], C2×C4×Dic5, C2×D10⋊C4 [×3], C10×C4⋊C4, C10.90(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C24.C22, C2×C4×D5, C4○D20, D4×D5, D42D5, Q82D5 [×2], C2×C5⋊D4, C4⋊C47D5, D208C4, D10.13D4, C4⋊C4⋊D5, C4×C5⋊D4, Dic5⋊D4, C20.23D4, C10.90(C4×D4)

Smallest permutation representation of C10.90(C4×D4)
On 160 points
Generators in S160
(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)
(1 138 50 153)(2 137 41 152)(3 136 42 151)(4 135 43 160)(5 134 44 159)(6 133 45 158)(7 132 46 157)(8 131 47 156)(9 140 48 155)(10 139 49 154)(11 30 150 38)(12 29 141 37)(13 28 142 36)(14 27 143 35)(15 26 144 34)(16 25 145 33)(17 24 146 32)(18 23 147 31)(19 22 148 40)(20 21 149 39)(51 115 76 100)(52 114 77 99)(53 113 78 98)(54 112 79 97)(55 111 80 96)(56 120 71 95)(57 119 72 94)(58 118 73 93)(59 117 74 92)(60 116 75 91)(61 129 86 104)(62 128 87 103)(63 127 88 102)(64 126 89 101)(65 125 90 110)(66 124 81 109)(67 123 82 108)(68 122 83 107)(69 121 84 106)(70 130 85 105)
(1 113 25 130)(2 114 26 121)(3 115 27 122)(4 116 28 123)(5 117 29 124)(6 118 30 125)(7 119 21 126)(8 120 22 127)(9 111 23 128)(10 112 24 129)(11 70 158 53)(12 61 159 54)(13 62 160 55)(14 63 151 56)(15 64 152 57)(16 65 153 58)(17 66 154 59)(18 67 155 60)(19 68 156 51)(20 69 157 52)(31 103 48 96)(32 104 49 97)(33 105 50 98)(34 106 41 99)(35 107 42 100)(36 108 43 91)(37 109 44 92)(38 110 45 93)(39 101 46 94)(40 102 47 95)(71 143 88 136)(72 144 89 137)(73 145 90 138)(74 146 81 139)(75 147 82 140)(76 148 83 131)(77 149 84 132)(78 150 85 133)(79 141 86 134)(80 142 87 135)
(1 110 6 105)(2 109 7 104)(3 108 8 103)(4 107 9 102)(5 106 10 101)(11 78 16 73)(12 77 17 72)(13 76 18 71)(14 75 19 80)(15 74 20 79)(21 97 26 92)(22 96 27 91)(23 95 28 100)(24 94 29 99)(25 93 30 98)(31 120 36 115)(32 119 37 114)(33 118 38 113)(34 117 39 112)(35 116 40 111)(41 124 46 129)(42 123 47 128)(43 122 48 127)(44 121 49 126)(45 130 50 125)(51 147 56 142)(52 146 57 141)(53 145 58 150)(54 144 59 149)(55 143 60 148)(61 137 66 132)(62 136 67 131)(63 135 68 140)(64 134 69 139)(65 133 70 138)(81 157 86 152)(82 156 87 151)(83 155 88 160)(84 154 89 159)(85 153 90 158)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4P4Q4R5A5B10A···10N20A···20X
order12···222444444444···4445510···1020···20
size11···120202222444410···102020222···24···4

68 irreducible representations

dim11111122222222444
type++++++++++-+
imageC1C2C2C2C2C4D4D4D5C4○D4D10C4×D5C5⋊D4C4○D20D4×D5D42D5Q82D5
kernelC10.90(C4×D4)C10.10C42C2×C4×Dic5C2×D10⋊C4C10×C4⋊C4D10⋊C4C2×Dic5C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps12131822286888224

Matrix representation of C10.90(C4×D4) in GL6(𝔽41)

770000
3500000
0013400
0073400
0000400
0000040
,
38370000
2330000
00141400
00302700
000098
0000032
,
2470000
35170000
00244000
0011700
0000320
0000109
,
340000
18380000
003300
00243800
000090
000009

G:=sub<GL(6,GF(41))| [7,35,0,0,0,0,7,0,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[38,23,0,0,0,0,37,3,0,0,0,0,0,0,14,30,0,0,0,0,14,27,0,0,0,0,0,0,9,0,0,0,0,0,8,32],[24,35,0,0,0,0,7,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,32,10,0,0,0,0,0,9],[3,18,0,0,0,0,4,38,0,0,0,0,0,0,3,24,0,0,0,0,3,38,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C10.90(C4×D4) in GAP, Magma, Sage, TeX

C_{10}._{90}(C_4\times D_4)
% in TeX

G:=Group("C10.90(C4xD4)");
// GroupNames label

G:=SmallGroup(320,617);
// by ID

G=gap.SmallGroup(320,617);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,758,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=a^5,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

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