Copied to
clipboard

G = C4×C56order 224 = 25·7

Abelian group of type [4,56]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C56, SmallGroup(224,45)

Series: Derived Chief Lower central Upper central

C1 — C4×C56
C1C2C22C2×C4C2×C28C2×C56 — C4×C56
C1 — C4×C56
C1 — C4×C56

Generators and relations for C4×C56
 G = < a,b | a4=b56=1, ab=ba >


Smallest permutation representation of C4×C56
Regular action on 224 points
Generators in S224
(1 179 79 156)(2 180 80 157)(3 181 81 158)(4 182 82 159)(5 183 83 160)(6 184 84 161)(7 185 85 162)(8 186 86 163)(9 187 87 164)(10 188 88 165)(11 189 89 166)(12 190 90 167)(13 191 91 168)(14 192 92 113)(15 193 93 114)(16 194 94 115)(17 195 95 116)(18 196 96 117)(19 197 97 118)(20 198 98 119)(21 199 99 120)(22 200 100 121)(23 201 101 122)(24 202 102 123)(25 203 103 124)(26 204 104 125)(27 205 105 126)(28 206 106 127)(29 207 107 128)(30 208 108 129)(31 209 109 130)(32 210 110 131)(33 211 111 132)(34 212 112 133)(35 213 57 134)(36 214 58 135)(37 215 59 136)(38 216 60 137)(39 217 61 138)(40 218 62 139)(41 219 63 140)(42 220 64 141)(43 221 65 142)(44 222 66 143)(45 223 67 144)(46 224 68 145)(47 169 69 146)(48 170 70 147)(49 171 71 148)(50 172 72 149)(51 173 73 150)(52 174 74 151)(53 175 75 152)(54 176 76 153)(55 177 77 154)(56 178 78 155)
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)

G:=sub<Sym(224)| (1,179,79,156)(2,180,80,157)(3,181,81,158)(4,182,82,159)(5,183,83,160)(6,184,84,161)(7,185,85,162)(8,186,86,163)(9,187,87,164)(10,188,88,165)(11,189,89,166)(12,190,90,167)(13,191,91,168)(14,192,92,113)(15,193,93,114)(16,194,94,115)(17,195,95,116)(18,196,96,117)(19,197,97,118)(20,198,98,119)(21,199,99,120)(22,200,100,121)(23,201,101,122)(24,202,102,123)(25,203,103,124)(26,204,104,125)(27,205,105,126)(28,206,106,127)(29,207,107,128)(30,208,108,129)(31,209,109,130)(32,210,110,131)(33,211,111,132)(34,212,112,133)(35,213,57,134)(36,214,58,135)(37,215,59,136)(38,216,60,137)(39,217,61,138)(40,218,62,139)(41,219,63,140)(42,220,64,141)(43,221,65,142)(44,222,66,143)(45,223,67,144)(46,224,68,145)(47,169,69,146)(48,170,70,147)(49,171,71,148)(50,172,72,149)(51,173,73,150)(52,174,74,151)(53,175,75,152)(54,176,76,153)(55,177,77,154)(56,178,78,155), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)>;

G:=Group( (1,179,79,156)(2,180,80,157)(3,181,81,158)(4,182,82,159)(5,183,83,160)(6,184,84,161)(7,185,85,162)(8,186,86,163)(9,187,87,164)(10,188,88,165)(11,189,89,166)(12,190,90,167)(13,191,91,168)(14,192,92,113)(15,193,93,114)(16,194,94,115)(17,195,95,116)(18,196,96,117)(19,197,97,118)(20,198,98,119)(21,199,99,120)(22,200,100,121)(23,201,101,122)(24,202,102,123)(25,203,103,124)(26,204,104,125)(27,205,105,126)(28,206,106,127)(29,207,107,128)(30,208,108,129)(31,209,109,130)(32,210,110,131)(33,211,111,132)(34,212,112,133)(35,213,57,134)(36,214,58,135)(37,215,59,136)(38,216,60,137)(39,217,61,138)(40,218,62,139)(41,219,63,140)(42,220,64,141)(43,221,65,142)(44,222,66,143)(45,223,67,144)(46,224,68,145)(47,169,69,146)(48,170,70,147)(49,171,71,148)(50,172,72,149)(51,173,73,150)(52,174,74,151)(53,175,75,152)(54,176,76,153)(55,177,77,154)(56,178,78,155), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224) );

G=PermutationGroup([(1,179,79,156),(2,180,80,157),(3,181,81,158),(4,182,82,159),(5,183,83,160),(6,184,84,161),(7,185,85,162),(8,186,86,163),(9,187,87,164),(10,188,88,165),(11,189,89,166),(12,190,90,167),(13,191,91,168),(14,192,92,113),(15,193,93,114),(16,194,94,115),(17,195,95,116),(18,196,96,117),(19,197,97,118),(20,198,98,119),(21,199,99,120),(22,200,100,121),(23,201,101,122),(24,202,102,123),(25,203,103,124),(26,204,104,125),(27,205,105,126),(28,206,106,127),(29,207,107,128),(30,208,108,129),(31,209,109,130),(32,210,110,131),(33,211,111,132),(34,212,112,133),(35,213,57,134),(36,214,58,135),(37,215,59,136),(38,216,60,137),(39,217,61,138),(40,218,62,139),(41,219,63,140),(42,220,64,141),(43,221,65,142),(44,222,66,143),(45,223,67,144),(46,224,68,145),(47,169,69,146),(48,170,70,147),(49,171,71,148),(50,172,72,149),(51,173,73,150),(52,174,74,151),(53,175,75,152),(54,176,76,153),(55,177,77,154),(56,178,78,155)], [(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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)])

C4×C56 is a maximal subgroup of
C42.279D14  C56⋊C8  C4.8Dic28  C562C8  C561C8  C4.17D56  C56.C8  C28⋊C16  C56.16Q8  C5611Q8  C569Q8  C28.14Q16  C568Q8  C56.13Q8  C42.282D14  C86D28  D14.C42  C42.243D14  C85D28  C4.5D56  C284D8  C8.8D28  C42.264D14  C284Q16  D5611C4

224 conjugacy classes

class 1 2A2B2C4A···4L7A···7F8A···8P14A···14R28A···28BT56A···56CR
order12224···47···78···814···1428···2856···56
size11111···11···11···11···11···11···1

224 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C7C8C14C14C28C28C56
kernelC4×C56C4×C28C2×C56C56C2×C28C4×C8C28C42C2×C8C8C2×C4C4
# reps11284616612482496

Matrix representation of C4×C56 in GL2(𝔽113) generated by

150
015
,
850
025
G:=sub<GL(2,GF(113))| [15,0,0,15],[85,0,0,25] >;

C4×C56 in GAP, Magma, Sage, TeX

C_4\times C_{56}
% in TeX

G:=Group("C4xC56");
// GroupNames label

G:=SmallGroup(224,45);
// by ID

G=gap.SmallGroup(224,45);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,168,343,117]);
// Polycyclic

G:=Group<a,b|a^4=b^56=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C56 in TeX

׿
×
𝔽