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G = C3⋊C64order 192 = 26·3

The semidirect product of C3 and C64 acting via C64/C32=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3⋊C64, C6.C32, C96.3C2, C24.2C8, C48.3C4, C32.2S3, C12.2C16, C16.3Dic3, C2.(C3⋊C32), C8.4(C3⋊C8), C4.2(C3⋊C16), SmallGroup(192,1)

Series: Derived Chief Lower central Upper central

C1C3 — C3⋊C64
C1C3C6C12C24C48C96 — C3⋊C64
C3 — C3⋊C64
C1C32

Generators and relations for C3⋊C64
 G = < a,b | a3=b64=1, bab-1=a-1 >

3C64

Smallest permutation representation of C3⋊C64
Regular action on 192 points
Generators in S192
(1 180 105)(2 106 181)(3 182 107)(4 108 183)(5 184 109)(6 110 185)(7 186 111)(8 112 187)(9 188 113)(10 114 189)(11 190 115)(12 116 191)(13 192 117)(14 118 129)(15 130 119)(16 120 131)(17 132 121)(18 122 133)(19 134 123)(20 124 135)(21 136 125)(22 126 137)(23 138 127)(24 128 139)(25 140 65)(26 66 141)(27 142 67)(28 68 143)(29 144 69)(30 70 145)(31 146 71)(32 72 147)(33 148 73)(34 74 149)(35 150 75)(36 76 151)(37 152 77)(38 78 153)(39 154 79)(40 80 155)(41 156 81)(42 82 157)(43 158 83)(44 84 159)(45 160 85)(46 86 161)(47 162 87)(48 88 163)(49 164 89)(50 90 165)(51 166 91)(52 92 167)(53 168 93)(54 94 169)(55 170 95)(56 96 171)(57 172 97)(58 98 173)(59 174 99)(60 100 175)(61 176 101)(62 102 177)(63 178 103)(64 104 179)
(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)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)

G:=sub<Sym(192)| (1,180,105)(2,106,181)(3,182,107)(4,108,183)(5,184,109)(6,110,185)(7,186,111)(8,112,187)(9,188,113)(10,114,189)(11,190,115)(12,116,191)(13,192,117)(14,118,129)(15,130,119)(16,120,131)(17,132,121)(18,122,133)(19,134,123)(20,124,135)(21,136,125)(22,126,137)(23,138,127)(24,128,139)(25,140,65)(26,66,141)(27,142,67)(28,68,143)(29,144,69)(30,70,145)(31,146,71)(32,72,147)(33,148,73)(34,74,149)(35,150,75)(36,76,151)(37,152,77)(38,78,153)(39,154,79)(40,80,155)(41,156,81)(42,82,157)(43,158,83)(44,84,159)(45,160,85)(46,86,161)(47,162,87)(48,88,163)(49,164,89)(50,90,165)(51,166,91)(52,92,167)(53,168,93)(54,94,169)(55,170,95)(56,96,171)(57,172,97)(58,98,173)(59,174,99)(60,100,175)(61,176,101)(62,102,177)(63,178,103)(64,104,179), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)>;

G:=Group( (1,180,105)(2,106,181)(3,182,107)(4,108,183)(5,184,109)(6,110,185)(7,186,111)(8,112,187)(9,188,113)(10,114,189)(11,190,115)(12,116,191)(13,192,117)(14,118,129)(15,130,119)(16,120,131)(17,132,121)(18,122,133)(19,134,123)(20,124,135)(21,136,125)(22,126,137)(23,138,127)(24,128,139)(25,140,65)(26,66,141)(27,142,67)(28,68,143)(29,144,69)(30,70,145)(31,146,71)(32,72,147)(33,148,73)(34,74,149)(35,150,75)(36,76,151)(37,152,77)(38,78,153)(39,154,79)(40,80,155)(41,156,81)(42,82,157)(43,158,83)(44,84,159)(45,160,85)(46,86,161)(47,162,87)(48,88,163)(49,164,89)(50,90,165)(51,166,91)(52,92,167)(53,168,93)(54,94,169)(55,170,95)(56,96,171)(57,172,97)(58,98,173)(59,174,99)(60,100,175)(61,176,101)(62,102,177)(63,178,103)(64,104,179), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192) );

G=PermutationGroup([(1,180,105),(2,106,181),(3,182,107),(4,108,183),(5,184,109),(6,110,185),(7,186,111),(8,112,187),(9,188,113),(10,114,189),(11,190,115),(12,116,191),(13,192,117),(14,118,129),(15,130,119),(16,120,131),(17,132,121),(18,122,133),(19,134,123),(20,124,135),(21,136,125),(22,126,137),(23,138,127),(24,128,139),(25,140,65),(26,66,141),(27,142,67),(28,68,143),(29,144,69),(30,70,145),(31,146,71),(32,72,147),(33,148,73),(34,74,149),(35,150,75),(36,76,151),(37,152,77),(38,78,153),(39,154,79),(40,80,155),(41,156,81),(42,82,157),(43,158,83),(44,84,159),(45,160,85),(46,86,161),(47,162,87),(48,88,163),(49,164,89),(50,90,165),(51,166,91),(52,92,167),(53,168,93),(54,94,169),(55,170,95),(56,96,171),(57,172,97),(58,98,173),(59,174,99),(60,100,175),(61,176,101),(62,102,177),(63,178,103),(64,104,179)], [(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),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)])

96 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D12A12B16A···16H24A24B24C24D32A···32P48A···48H64A···64AF96A···96P
order1234468888121216···162424242432···3248···4864···6496···96
size1121121111221···122221···12···23···32···2

96 irreducible representations

dim1111111222222
type+++-
imageC1C2C4C8C16C32C64S3Dic3C3⋊C8C3⋊C16C3⋊C32C3⋊C64
kernelC3⋊C64C96C48C24C12C6C3C32C16C8C4C2C1
# reps1124816321124816

Matrix representation of C3⋊C64 in GL3(𝔽193) generated by

100
001
0192192
,
7400
018640
0477
G:=sub<GL(3,GF(193))| [1,0,0,0,0,192,0,1,192],[74,0,0,0,186,47,0,40,7] >;

C3⋊C64 in GAP, Magma, Sage, TeX

C_3\rtimes C_{64}
% in TeX

G:=Group("C3:C64");
// GroupNames label

G:=SmallGroup(192,1);
// by ID

G=gap.SmallGroup(192,1);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,14,36,58,80,102,6278]);
// Polycyclic

G:=Group<a,b|a^3=b^64=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C3⋊C64 in TeX

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