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G = C20.64(C4⋊C4)  order 320 = 26·5

11st non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.64(C4⋊C4), C20.64(C2×Q8), (C2×C20).21Q8, C4⋊C4.232D10, (C2×C20).471D4, C4.Dic514C4, C10.D827C2, C20.Q827C2, (C2×C4).14Dic10, C4.29(C2×Dic10), C42⋊C2.4D5, (C22×C10).71D4, C2.1(D4⋊D10), C55(M4(2)⋊C4), (C2×C20).325C23, C20.121(C22×C4), (C22×C4).107D10, C23.53(C5⋊D4), C10.104(C8⋊C22), C2.1(D4.9D10), C4.25(C10.D4), C4⋊Dic5.324C22, C10.104(C8.C22), (C22×C20).145C22, C22.8(C10.D4), C4.87(C2×C4×D5), C52C87(C2×C4), C10.62(C2×C4⋊C4), (C2×C4).43(C4×D5), (C2×C10).39(C4⋊C4), (C2×C20).260(C2×C4), (C2×C10).454(C2×D4), (C2×C4⋊Dic5).37C2, C22.70(C2×C5⋊D4), (C2×C4).240(C5⋊D4), (C5×C4⋊C4).263C22, (C5×C42⋊C2).4C2, (C2×C52C8).85C22, C2.13(C2×C10.D4), (C2×C4).425(C22×D5), (C2×C4.Dic5).16C2, SmallGroup(320,622)

Series: Derived Chief Lower central Upper central

C1C20 — C20.64(C4⋊C4)
C1C5C10C20C2×C20C4⋊Dic5C2×C4⋊Dic5 — C20.64(C4⋊C4)
C5C10C20 — C20.64(C4⋊C4)
C1C22C22×C4C42⋊C2

Generators and relations for C20.64(C4⋊C4)
 G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, ac=ca, cbc-1=a10b3 >

Subgroups: 350 in 118 conjugacy classes, 63 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×C10, M4(2)⋊C4, C2×C52C8 [×2], C4.Dic5 [×4], C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C22×Dic5, C22×C20, C10.D8 [×2], C20.Q8 [×2], C2×C4.Dic5, C2×C4⋊Dic5, C5×C42⋊C2, C20.64(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8⋊C22, C8.C22, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, M4(2)⋊C4, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4, D4⋊D10, D4.9D10, C20.64(C4⋊C4)

Smallest permutation representation of C20.64(C4⋊C4)
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)
(1 139 117 86 11 129 107 96)(2 138 118 85 12 128 108 95)(3 137 119 84 13 127 109 94)(4 136 120 83 14 126 110 93)(5 135 101 82 15 125 111 92)(6 134 102 81 16 124 112 91)(7 133 103 100 17 123 113 90)(8 132 104 99 18 122 114 89)(9 131 105 98 19 121 115 88)(10 130 106 97 20 140 116 87)(21 45 144 80 31 55 154 70)(22 44 145 79 32 54 155 69)(23 43 146 78 33 53 156 68)(24 42 147 77 34 52 157 67)(25 41 148 76 35 51 158 66)(26 60 149 75 36 50 159 65)(27 59 150 74 37 49 160 64)(28 58 151 73 38 48 141 63)(29 57 152 72 39 47 142 62)(30 56 153 71 40 46 143 61)
(1 32 102 150)(2 33 103 151)(3 34 104 152)(4 35 105 153)(5 36 106 154)(6 37 107 155)(7 38 108 156)(8 39 109 157)(9 40 110 158)(10 21 111 159)(11 22 112 160)(12 23 113 141)(13 24 114 142)(14 25 115 143)(15 26 116 144)(16 27 117 145)(17 28 118 146)(18 29 119 147)(19 30 120 148)(20 31 101 149)(41 121 61 83)(42 122 62 84)(43 123 63 85)(44 124 64 86)(45 125 65 87)(46 126 66 88)(47 127 67 89)(48 128 68 90)(49 129 69 91)(50 130 70 92)(51 131 71 93)(52 132 72 94)(53 133 73 95)(54 134 74 96)(55 135 75 97)(56 136 76 98)(57 137 77 99)(58 138 78 100)(59 139 79 81)(60 140 80 82)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20AB
order12222244444444444455888810···101010101020···2020···20
size111122222244442020202022202020202···244442···24···4

62 irreducible representations

dim111111122222222224444
type+++++++-++++-+-+-
imageC1C2C2C2C2C2C4D4Q8D4D5D10D10Dic10C4×D5C5⋊D4C5⋊D4C8⋊C22C8.C22D4⋊D10D4.9D10
kernelC20.64(C4⋊C4)C10.D8C20.Q8C2×C4.Dic5C2×C4⋊Dic5C5×C42⋊C2C4.Dic5C2×C20C2×C20C22×C10C42⋊C2C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C10C10C2C2
# reps122111812124288441144

Matrix representation of C20.64(C4⋊C4) in GL6(𝔽41)

010000
4070000
00321100
00302700
00003211
00003027
,
0320000
3200000
0000919
0000032
0017300
00402400
,
11320000
9300000
000010
000001
0040000
0004000

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,7,0,0,0,0,0,0,32,30,0,0,0,0,11,27,0,0,0,0,0,0,32,30,0,0,0,0,11,27],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,0,17,40,0,0,0,0,3,24,0,0,9,0,0,0,0,0,19,32,0,0],[11,9,0,0,0,0,32,30,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C20.64(C4⋊C4) in GAP, Magma, Sage, TeX

C_{20}._{64}(C_4\rtimes C_4)
% in TeX

G:=Group("C20.64(C4:C4)");
// GroupNames label

G:=SmallGroup(320,622);
// by ID

G=gap.SmallGroup(320,622);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,387,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^10*b^3>;
// generators/relations

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