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G = C10×C8○D4order 320 = 26·5

Direct product of C10 and C8○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C10×C8○D4, C20.93C24, C40.80C23, C4○D4.4C20, D4.8(C2×C20), Q8.9(C2×C20), (C2×C40)⋊54C22, (C22×C40)⋊27C2, (C22×C8)⋊13C10, (D4×C10).37C4, (C2×D4).12C20, (Q8×C10).30C4, (C2×Q8).10C20, C4.17(C23×C10), C2.11(C23×C20), C4.22(C22×C20), C10.84(C23×C4), C23.20(C2×C20), C8.17(C22×C10), M4(2)⋊11(C2×C10), (C10×M4(2))⋊35C2, (C2×M4(2))⋊17C10, C20.226(C22×C4), (C2×C20).969C23, C22.4(C22×C20), (C5×M4(2))⋊40C22, (C22×C20).600C22, (C2×C8)⋊16(C2×C10), (C2×C4).53(C2×C20), (C5×C4○D4).12C4, (C5×D4).44(C2×C4), (C5×Q8).48(C2×C4), (C2×C20).447(C2×C4), (C10×C4○D4).28C2, (C2×C4○D4).14C10, C4○D4.14(C2×C10), (C5×C4○D4).59C22, (C22×C4).127(C2×C10), (C22×C10).154(C2×C4), (C2×C10).136(C22×C4), (C2×C4).139(C22×C10), SmallGroup(320,1569)

Series: Derived Chief Lower central Upper central

C1C2 — C10×C8○D4
C1C2C4C20C40C2×C40C5×C8○D4 — C10×C8○D4
C1C2 — C10×C8○D4
C1C2×C40 — C10×C8○D4

Generators and relations for C10×C8○D4
 G = < a,b,c,d | a10=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 290 in 266 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C40 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C2×C8○D4, C2×C40, C2×C40 [×15], C5×M4(2) [×12], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C22×C40 [×3], C10×M4(2) [×3], C5×C8○D4 [×8], C10×C4○D4, C10×C8○D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C8○D4 [×2], C23×C4, C2×C20 [×28], C22×C10 [×15], C2×C8○D4, C22×C20 [×14], C23×C10, C5×C8○D4 [×2], C23×C20, C10×C8○D4

Smallest permutation representation of C10×C8○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)
(1 125 62 133 52 147 50 119)(2 126 63 134 53 148 41 120)(3 127 64 135 54 149 42 111)(4 128 65 136 55 150 43 112)(5 129 66 137 56 141 44 113)(6 130 67 138 57 142 45 114)(7 121 68 139 58 143 46 115)(8 122 69 140 59 144 47 116)(9 123 70 131 60 145 48 117)(10 124 61 132 51 146 49 118)(11 78 31 84 153 92 22 106)(12 79 32 85 154 93 23 107)(13 80 33 86 155 94 24 108)(14 71 34 87 156 95 25 109)(15 72 35 88 157 96 26 110)(16 73 36 89 158 97 27 101)(17 74 37 90 159 98 28 102)(18 75 38 81 160 99 29 103)(19 76 39 82 151 100 30 104)(20 77 40 83 152 91 21 105)
(1 74 52 98)(2 75 53 99)(3 76 54 100)(4 77 55 91)(5 78 56 92)(6 79 57 93)(7 80 58 94)(8 71 59 95)(9 72 60 96)(10 73 51 97)(11 137 153 113)(12 138 154 114)(13 139 155 115)(14 140 156 116)(15 131 157 117)(16 132 158 118)(17 133 159 119)(18 134 160 120)(19 135 151 111)(20 136 152 112)(21 128 40 150)(22 129 31 141)(23 130 32 142)(24 121 33 143)(25 122 34 144)(26 123 35 145)(27 124 36 146)(28 125 37 147)(29 126 38 148)(30 127 39 149)(41 103 63 81)(42 104 64 82)(43 105 65 83)(44 106 66 84)(45 107 67 85)(46 108 68 86)(47 109 69 87)(48 110 70 88)(49 101 61 89)(50 102 62 90)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 118)(12 119)(13 120)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 141)(28 142)(29 143)(30 144)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 121)(39 122)(40 123)(41 108)(42 109)(43 110)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 91)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 81)(69 82)(70 83)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 151)

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

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

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

200 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J5A5B5C5D8A···8H8I···8T10A···10L10M···10AJ20A···20P20Q···20AN40A···40AF40AG···40CB
order12222···244444···455558···88···810···1010···1020···2020···2040···4040···40
size11112···211112···211111···12···21···12···21···12···21···12···2

200 irreducible representations

dim111111111111111122
type+++++
imageC1C2C2C2C2C4C4C4C5C10C10C10C10C20C20C20C8○D4C5×C8○D4
kernelC10×C8○D4C22×C40C10×M4(2)C5×C8○D4C10×C4○D4D4×C10Q8×C10C5×C4○D4C2×C8○D4C22×C8C2×M4(2)C8○D4C2×C4○D4C2×D4C2×Q8C4○D4C10C2
# reps133816284121232424832832

Matrix representation of C10×C8○D4 in GL3(𝔽41) generated by

4000
0160
0016
,
100
0140
0014
,
100
001
0400
,
100
0040
0400
G:=sub<GL(3,GF(41))| [40,0,0,0,16,0,0,0,16],[1,0,0,0,14,0,0,0,14],[1,0,0,0,0,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;

C10×C8○D4 in GAP, Magma, Sage, TeX

C_{10}\times C_8\circ D_4
% in TeX

G:=Group("C10xC8oD4");
// GroupNames label

G:=SmallGroup(320,1569);
// by ID

G=gap.SmallGroup(320,1569);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1731,124]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

׿
×
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