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G = C20.(C4○D4)  order 320 = 26·5

32nd non-split extension by C20 of C4○D4 acting via C4○D4/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Q810D5, C4⋊C4.41D10, (C2×C8).140D10, C20.32(C4○D4), C10.57(C4○D8), C4.75(C4○D20), C10.Q1617C2, C10.D817C2, D102Q8.6C2, (C22×D5).34D4, C22.219(D4×D5), D101C8.14C2, C20.44D432C2, (C2×C40).287C22, (C2×C20).283C23, C4.27(Q82D5), (C2×Dic5).220D4, C54(C23.20D4), C2.24(SD16⋊D5), C10.43(C8.C22), C4⋊Dic5.113C22, C2.24(SD163D5), (C2×Dic10).88C22, C2.14(D10.13D4), C10.44(C22.D4), (C5×C4.Q8)⋊18C2, C4⋊C47D5.6C2, (C2×C4×D5).39C22, (C2×C10).288(C2×D4), (C5×C4⋊C4).76C22, (C2×C52C8).60C22, (C2×C4).386(C22×D5), SmallGroup(320,494)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.(C4○D4)
C1C5C10C2×C10C2×C20C2×C4×D5C4⋊C47D5 — C20.(C4○D4)
C5C10C2×C20 — C20.(C4○D4)
C1C22C2×C4C4.Q8

Generators and relations for C20.(C4○D4)
 G = < a,b,c,d | a10=b4=1, c4=b2, d2=a5, bab-1=cac-1=dad-1=a-1, cbc-1=a5b, bd=db, dcd-1=b2c3 >

Subgroups: 382 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C22⋊C8, Q8⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C40, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.20D4, C2×C52C8, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C10.D8, C10.Q16, C20.44D4, D101C8, C5×C4.Q8, C4⋊C47D5, D102Q8, C20.(C4○D4)
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8.C22, C22×D5, C23.20D4, C4○D20, D4×D5, Q82D5, D10.13D4, SD16⋊D5, SD163D5, C20.(C4○D4)

Smallest permutation representation of C20.(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 31 11 45)(2 40 12 44)(3 39 13 43)(4 38 14 42)(5 37 15 41)(6 36 16 50)(7 35 17 49)(8 34 18 48)(9 33 19 47)(10 32 20 46)(21 158 140 143)(22 157 131 142)(23 156 132 141)(24 155 133 150)(25 154 134 149)(26 153 135 148)(27 152 136 147)(28 151 137 146)(29 160 138 145)(30 159 139 144)(51 62 86 77)(52 61 87 76)(53 70 88 75)(54 69 89 74)(55 68 90 73)(56 67 81 72)(57 66 82 71)(58 65 83 80)(59 64 84 79)(60 63 85 78)(91 107 126 112)(92 106 127 111)(93 105 128 120)(94 104 129 119)(95 103 130 118)(96 102 121 117)(97 101 122 116)(98 110 123 115)(99 109 124 114)(100 108 125 113)
(1 60 36 63 11 85 50 78)(2 59 37 62 12 84 41 77)(3 58 38 61 13 83 42 76)(4 57 39 70 14 82 43 75)(5 56 40 69 15 81 44 74)(6 55 31 68 16 90 45 73)(7 54 32 67 17 89 46 72)(8 53 33 66 18 88 47 71)(9 52 34 65 19 87 48 80)(10 51 35 64 20 86 49 79)(21 103 143 100 140 118 158 125)(22 102 144 99 131 117 159 124)(23 101 145 98 132 116 160 123)(24 110 146 97 133 115 151 122)(25 109 147 96 134 114 152 121)(26 108 148 95 135 113 153 130)(27 107 149 94 136 112 154 129)(28 106 150 93 137 111 155 128)(29 105 141 92 138 120 156 127)(30 104 142 91 139 119 157 126)
(1 118 6 113)(2 117 7 112)(3 116 8 111)(4 115 9 120)(5 114 10 119)(11 103 16 108)(12 102 17 107)(13 101 18 106)(14 110 19 105)(15 109 20 104)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 95 36 100)(32 94 37 99)(33 93 38 98)(34 92 39 97)(35 91 40 96)(41 124 46 129)(42 123 47 128)(43 122 48 127)(44 121 49 126)(45 130 50 125)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 145 66 150)(62 144 67 149)(63 143 68 148)(64 142 69 147)(65 141 70 146)(71 155 76 160)(72 154 77 159)(73 153 78 158)(74 152 79 157)(75 151 80 156)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444444455888810···102020202020···2040···40
size111120224481010202040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8C4○D20C8.C22Q82D5D4×D5SD16⋊D5SD163D5
kernelC20.(C4○D4)C10.D8C10.Q16C20.44D4D101C8C5×C4.Q8C4⋊C47D5D102Q8C2×Dic5C22×D5C4.Q8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111111124424812244

Matrix representation of C20.(C4○D4) in GL4(𝔽41) generated by

13400
73400
0010
0001
,
13400
04000
0090
0009
,
243800
11700
00380
001627
,
91900
03200
00305
001711
G:=sub<GL(4,GF(41))| [1,7,0,0,34,34,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,40,0,0,0,0,9,0,0,0,0,9],[24,1,0,0,38,17,0,0,0,0,38,16,0,0,0,27],[9,0,0,0,19,32,0,0,0,0,30,17,0,0,5,11] >;

C20.(C4○D4) in GAP, Magma, Sage, TeX

C_{20}.(C_4\circ D_4)
% in TeX

G:=Group("C20.(C4oD4)");
// GroupNames label

G:=SmallGroup(320,494);
// by ID

G=gap.SmallGroup(320,494);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,926,219,100,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=b^2*c^3>;
// generators/relations

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