metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊4C4, C8⋊3Dic7, C14.4C42, C14.2M4(2), C7⋊C8⋊4C4, (C2×C8).8D7, C7⋊2(C8⋊C4), C4.21(C4×D7), C28.41(C2×C4), (C2×C56).12C2, (C2×C4).92D14, C2.4(C4×Dic7), C2.2(C8⋊D7), (C4×Dic7).6C2, (C2×Dic7).3C4, C4.13(C2×Dic7), C22.10(C4×D7), (C2×C28).106C22, (C2×C7⋊C8).10C2, (C2×C14).11(C2×C4), SmallGroup(224,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊C4
G = < a,b | a56=b4=1, bab-1=a13 >
Smallest permutation representation of C56⋊C4
►Regular action on 224 points
Generators in S224
(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)
(1 155 61 182)(2 168 62 195)(3 125 63 208)(4 138 64 221)(5 151 65 178)(6 164 66 191)(7 121 67 204)(8 134 68 217)(9 147 69 174)(10 160 70 187)(11 117 71 200)(12 130 72 213)(13 143 73 170)(14 156 74 183)(15 113 75 196)(16 126 76 209)(17 139 77 222)(18 152 78 179)(19 165 79 192)(20 122 80 205)(21 135 81 218)(22 148 82 175)(23 161 83 188)(24 118 84 201)(25 131 85 214)(26 144 86 171)(27 157 87 184)(28 114 88 197)(29 127 89 210)(30 140 90 223)(31 153 91 180)(32 166 92 193)(33 123 93 206)(34 136 94 219)(35 149 95 176)(36 162 96 189)(37 119 97 202)(38 132 98 215)(39 145 99 172)(40 158 100 185)(41 115 101 198)(42 128 102 211)(43 141 103 224)(44 154 104 181)(45 167 105 194)(46 124 106 207)(47 137 107 220)(48 150 108 177)(49 163 109 190)(50 120 110 203)(51 133 111 216)(52 146 112 173)(53 159 57 186)(54 116 58 199)(55 129 59 212)(56 142 60 169)
G:=sub<Sym(224)| (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), (1,155,61,182)(2,168,62,195)(3,125,63,208)(4,138,64,221)(5,151,65,178)(6,164,66,191)(7,121,67,204)(8,134,68,217)(9,147,69,174)(10,160,70,187)(11,117,71,200)(12,130,72,213)(13,143,73,170)(14,156,74,183)(15,113,75,196)(16,126,76,209)(17,139,77,222)(18,152,78,179)(19,165,79,192)(20,122,80,205)(21,135,81,218)(22,148,82,175)(23,161,83,188)(24,118,84,201)(25,131,85,214)(26,144,86,171)(27,157,87,184)(28,114,88,197)(29,127,89,210)(30,140,90,223)(31,153,91,180)(32,166,92,193)(33,123,93,206)(34,136,94,219)(35,149,95,176)(36,162,96,189)(37,119,97,202)(38,132,98,215)(39,145,99,172)(40,158,100,185)(41,115,101,198)(42,128,102,211)(43,141,103,224)(44,154,104,181)(45,167,105,194)(46,124,106,207)(47,137,107,220)(48,150,108,177)(49,163,109,190)(50,120,110,203)(51,133,111,216)(52,146,112,173)(53,159,57,186)(54,116,58,199)(55,129,59,212)(56,142,60,169)>;
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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(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), (1,155,61,182)(2,168,62,195)(3,125,63,208)(4,138,64,221)(5,151,65,178)(6,164,66,191)(7,121,67,204)(8,134,68,217)(9,147,69,174)(10,160,70,187)(11,117,71,200)(12,130,72,213)(13,143,73,170)(14,156,74,183)(15,113,75,196)(16,126,76,209)(17,139,77,222)(18,152,78,179)(19,165,79,192)(20,122,80,205)(21,135,81,218)(22,148,82,175)(23,161,83,188)(24,118,84,201)(25,131,85,214)(26,144,86,171)(27,157,87,184)(28,114,88,197)(29,127,89,210)(30,140,90,223)(31,153,91,180)(32,166,92,193)(33,123,93,206)(34,136,94,219)(35,149,95,176)(36,162,96,189)(37,119,97,202)(38,132,98,215)(39,145,99,172)(40,158,100,185)(41,115,101,198)(42,128,102,211)(43,141,103,224)(44,154,104,181)(45,167,105,194)(46,124,106,207)(47,137,107,220)(48,150,108,177)(49,163,109,190)(50,120,110,203)(51,133,111,216)(52,146,112,173)(53,159,57,186)(54,116,58,199)(55,129,59,212)(56,142,60,169) );
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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(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)], [(1,155,61,182),(2,168,62,195),(3,125,63,208),(4,138,64,221),(5,151,65,178),(6,164,66,191),(7,121,67,204),(8,134,68,217),(9,147,69,174),(10,160,70,187),(11,117,71,200),(12,130,72,213),(13,143,73,170),(14,156,74,183),(15,113,75,196),(16,126,76,209),(17,139,77,222),(18,152,78,179),(19,165,79,192),(20,122,80,205),(21,135,81,218),(22,148,82,175),(23,161,83,188),(24,118,84,201),(25,131,85,214),(26,144,86,171),(27,157,87,184),(28,114,88,197),(29,127,89,210),(30,140,90,223),(31,153,91,180),(32,166,92,193),(33,123,93,206),(34,136,94,219),(35,149,95,176),(36,162,96,189),(37,119,97,202),(38,132,98,215),(39,145,99,172),(40,158,100,185),(41,115,101,198),(42,128,102,211),(43,141,103,224),(44,154,104,181),(45,167,105,194),(46,124,106,207),(47,137,107,220),(48,150,108,177),(49,163,109,190),(50,120,110,203),(51,133,111,216),(52,146,112,173),(53,159,57,186),(54,116,58,199),(55,129,59,212),(56,142,60,169)]])
C56⋊C4 is a maximal subgroup of
C112⋊C4 C56⋊11Q8 C4×C8⋊D7 D14.C42 C56⋊Q8 D7×C8⋊C4 D14.4C42 Dic7.M4(2) C56⋊C4⋊C2 D14⋊2M4(2) C7⋊C8⋊26D4 D4.D7⋊C4 C4⋊C4.D14 C28⋊Q8⋊C2 D4⋊D7⋊C4 C7⋊Q16⋊C4 Q8⋊C4⋊D7 C56⋊C4.C2 Q8⋊D7⋊C4 C42.27D14 C28.M4(2) C42.202D14 C42.31D14 Dic28⋊9C4 C56⋊3Q8 D56⋊9C4 C56⋊4Q8 C56⋊C2⋊C4 D56⋊10C4 C28.12C42 C56⋊32D4 M4(2)×Dic7 C28.7C42 C56⋊D4 C56.50D4 D8⋊Dic7 C56⋊11D4 SD16⋊Dic7 C56.31D4 C56⋊9D4 Q16⋊Dic7 C56.37D4 D8⋊4Dic7
C56⋊C4 is a maximal quotient of
C42.279D14 C56⋊C8 C112⋊C4 (C2×C56)⋊5C4
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | M4(2) | Dic7 | D14 | C4×D7 | C4×D7 | C8⋊D7 |
| kernel | C56⋊C4 | C2×C7⋊C8 | C4×Dic7 | C2×C56 | C7⋊C8 | C56 | C2×Dic7 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 4 | 6 | 3 | 6 | 6 | 24 |
Matrix representation of C56⋊C4 ►in GL4(𝔽113) generated by
| 22 | 98 | 0 | 0 |
| 30 | 36 | 0 | 0 |
| 0 | 0 | 6 | 86 |
| 0 | 0 | 27 | 102 |
| 98 | 106 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 24 | 49 |
| 0 | 0 | 39 | 89 |
G:=sub<GL(4,GF(113))| [22,30,0,0,98,36,0,0,0,0,6,27,0,0,86,102],[98,0,0,0,106,15,0,0,0,0,24,39,0,0,49,89] >;
C56⋊C4 in GAP, Magma, Sage, TeX
C_{56}\rtimes C_4 % in TeX
G:=Group("C56:C4"); // GroupNames label
G:=SmallGroup(224,21);
// by ID
G=gap.SmallGroup(224,21);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,217,55,69,6917]);
// Polycyclic
G:=Group<a,b|a^56=b^4=1,b*a*b^-1=a^13>;
// generators/relations
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