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G = C4×C48order 192 = 26·3

Abelian group of type [4,48]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C48, SmallGroup(192,151)

Series: Derived Chief Lower central Upper central

C1 — C4×C48
C1C2C4C2×C4C2×C8C2×C24C2×C48 — C4×C48
C1 — C4×C48
C1 — C4×C48

Generators and relations for C4×C48
 G = < a,b | a4=b48=1, ab=ba >


Smallest permutation representation of C4×C48
Regular action on 192 points
Generators in S192
(1 50 189 131)(2 51 190 132)(3 52 191 133)(4 53 192 134)(5 54 145 135)(6 55 146 136)(7 56 147 137)(8 57 148 138)(9 58 149 139)(10 59 150 140)(11 60 151 141)(12 61 152 142)(13 62 153 143)(14 63 154 144)(15 64 155 97)(16 65 156 98)(17 66 157 99)(18 67 158 100)(19 68 159 101)(20 69 160 102)(21 70 161 103)(22 71 162 104)(23 72 163 105)(24 73 164 106)(25 74 165 107)(26 75 166 108)(27 76 167 109)(28 77 168 110)(29 78 169 111)(30 79 170 112)(31 80 171 113)(32 81 172 114)(33 82 173 115)(34 83 174 116)(35 84 175 117)(36 85 176 118)(37 86 177 119)(38 87 178 120)(39 88 179 121)(40 89 180 122)(41 90 181 123)(42 91 182 124)(43 92 183 125)(44 93 184 126)(45 94 185 127)(46 95 186 128)(47 96 187 129)(48 49 188 130)
(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)| (1,50,189,131)(2,51,190,132)(3,52,191,133)(4,53,192,134)(5,54,145,135)(6,55,146,136)(7,56,147,137)(8,57,148,138)(9,58,149,139)(10,59,150,140)(11,60,151,141)(12,61,152,142)(13,62,153,143)(14,63,154,144)(15,64,155,97)(16,65,156,98)(17,66,157,99)(18,67,158,100)(19,68,159,101)(20,69,160,102)(21,70,161,103)(22,71,162,104)(23,72,163,105)(24,73,164,106)(25,74,165,107)(26,75,166,108)(27,76,167,109)(28,77,168,110)(29,78,169,111)(30,79,170,112)(31,80,171,113)(32,81,172,114)(33,82,173,115)(34,83,174,116)(35,84,175,117)(36,85,176,118)(37,86,177,119)(38,87,178,120)(39,88,179,121)(40,89,180,122)(41,90,181,123)(42,91,182,124)(43,92,183,125)(44,93,184,126)(45,94,185,127)(46,95,186,128)(47,96,187,129)(48,49,188,130), (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( (1,50,189,131)(2,51,190,132)(3,52,191,133)(4,53,192,134)(5,54,145,135)(6,55,146,136)(7,56,147,137)(8,57,148,138)(9,58,149,139)(10,59,150,140)(11,60,151,141)(12,61,152,142)(13,62,153,143)(14,63,154,144)(15,64,155,97)(16,65,156,98)(17,66,157,99)(18,67,158,100)(19,68,159,101)(20,69,160,102)(21,70,161,103)(22,71,162,104)(23,72,163,105)(24,73,164,106)(25,74,165,107)(26,75,166,108)(27,76,167,109)(28,77,168,110)(29,78,169,111)(30,79,170,112)(31,80,171,113)(32,81,172,114)(33,82,173,115)(34,83,174,116)(35,84,175,117)(36,85,176,118)(37,86,177,119)(38,87,178,120)(39,88,179,121)(40,89,180,122)(41,90,181,123)(42,91,182,124)(43,92,183,125)(44,93,184,126)(45,94,185,127)(46,95,186,128)(47,96,187,129)(48,49,188,130), (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([(1,50,189,131),(2,51,190,132),(3,52,191,133),(4,53,192,134),(5,54,145,135),(6,55,146,136),(7,56,147,137),(8,57,148,138),(9,58,149,139),(10,59,150,140),(11,60,151,141),(12,61,152,142),(13,62,153,143),(14,63,154,144),(15,64,155,97),(16,65,156,98),(17,66,157,99),(18,67,158,100),(19,68,159,101),(20,69,160,102),(21,70,161,103),(22,71,162,104),(23,72,163,105),(24,73,164,106),(25,74,165,107),(26,75,166,108),(27,76,167,109),(28,77,168,110),(29,78,169,111),(30,79,170,112),(31,80,171,113),(32,81,172,114),(33,82,173,115),(34,83,174,116),(35,84,175,117),(36,85,176,118),(37,86,177,119),(38,87,178,120),(39,88,179,121),(40,89,180,122),(41,90,181,123),(42,91,182,124),(43,92,183,125),(44,93,184,126),(45,94,185,127),(46,95,186,128),(47,96,187,129),(48,49,188,130)], [(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)])

192 conjugacy classes

class 1 2A2B2C3A3B4A···4L6A···6F8A···8P12A···12X16A···16AF24A···24AF48A···48BL
order1222334···46···68···812···1216···1624···2448···48
size1111111···11···11···11···11···11···11···1

192 irreducible representations

dim111111111111111111
type+++
imageC1C2C2C3C4C4C4C6C6C8C8C12C12C12C16C24C24C48
kernelC4×C48C4×C24C2×C48C4×C16C48C4×C12C2×C24C4×C8C2×C16C24C2×C12C16C42C2×C8C12C8C2×C4C4
# reps11228222488164432161664

Matrix representation of C4×C48 in GL3(𝔽97) generated by

2200
010
0022
,
7000
0240
0033
G:=sub<GL(3,GF(97))| [22,0,0,0,1,0,0,0,22],[70,0,0,0,24,0,0,0,33] >;

C4×C48 in GAP, Magma, Sage, TeX

C_4\times C_{48}
% in TeX

G:=Group("C4xC48");
// GroupNames label

G:=SmallGroup(192,151);
// by ID

G=gap.SmallGroup(192,151);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,176,136,124]);
// Polycyclic

G:=Group<a,b|a^4=b^48=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C48 in TeX

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