metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊1C8, C28.7Q8, C28.32D4, C4.16D28, C42.2D7, C4.7Dic14, C14.5M4(2), C4⋊(C7⋊C8), C7⋊1(C4⋊C8), C14.7(C2×C8), (C4×C28).4C2, (C2×C28).7C4, C14.1(C4⋊C4), (C2×C4).89D14, (C2×C4).3Dic7, C2.1(C4⋊Dic7), C2.2(C4.Dic7), C22.8(C2×Dic7), (C2×C28).103C22, C2.3(C2×C7⋊C8), (C2×C7⋊C8).8C2, (C2×C14).26(C2×C4), SmallGroup(224,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊C8
G = < a,b | a28=b8=1, bab-1=a-1 >
Smallest permutation representation of C28⋊C8
►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 51 83 182 145 122 201 85)(2 50 84 181 146 121 202 112)(3 49 57 180 147 120 203 111)(4 48 58 179 148 119 204 110)(5 47 59 178 149 118 205 109)(6 46 60 177 150 117 206 108)(7 45 61 176 151 116 207 107)(8 44 62 175 152 115 208 106)(9 43 63 174 153 114 209 105)(10 42 64 173 154 113 210 104)(11 41 65 172 155 140 211 103)(12 40 66 171 156 139 212 102)(13 39 67 170 157 138 213 101)(14 38 68 169 158 137 214 100)(15 37 69 196 159 136 215 99)(16 36 70 195 160 135 216 98)(17 35 71 194 161 134 217 97)(18 34 72 193 162 133 218 96)(19 33 73 192 163 132 219 95)(20 32 74 191 164 131 220 94)(21 31 75 190 165 130 221 93)(22 30 76 189 166 129 222 92)(23 29 77 188 167 128 223 91)(24 56 78 187 168 127 224 90)(25 55 79 186 141 126 197 89)(26 54 80 185 142 125 198 88)(27 53 81 184 143 124 199 87)(28 52 82 183 144 123 200 86)
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), (1,51,83,182,145,122,201,85)(2,50,84,181,146,121,202,112)(3,49,57,180,147,120,203,111)(4,48,58,179,148,119,204,110)(5,47,59,178,149,118,205,109)(6,46,60,177,150,117,206,108)(7,45,61,176,151,116,207,107)(8,44,62,175,152,115,208,106)(9,43,63,174,153,114,209,105)(10,42,64,173,154,113,210,104)(11,41,65,172,155,140,211,103)(12,40,66,171,156,139,212,102)(13,39,67,170,157,138,213,101)(14,38,68,169,158,137,214,100)(15,37,69,196,159,136,215,99)(16,36,70,195,160,135,216,98)(17,35,71,194,161,134,217,97)(18,34,72,193,162,133,218,96)(19,33,73,192,163,132,219,95)(20,32,74,191,164,131,220,94)(21,31,75,190,165,130,221,93)(22,30,76,189,166,129,222,92)(23,29,77,188,167,128,223,91)(24,56,78,187,168,127,224,90)(25,55,79,186,141,126,197,89)(26,54,80,185,142,125,198,88)(27,53,81,184,143,124,199,87)(28,52,82,183,144,123,200,86)>;
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), (1,51,83,182,145,122,201,85)(2,50,84,181,146,121,202,112)(3,49,57,180,147,120,203,111)(4,48,58,179,148,119,204,110)(5,47,59,178,149,118,205,109)(6,46,60,177,150,117,206,108)(7,45,61,176,151,116,207,107)(8,44,62,175,152,115,208,106)(9,43,63,174,153,114,209,105)(10,42,64,173,154,113,210,104)(11,41,65,172,155,140,211,103)(12,40,66,171,156,139,212,102)(13,39,67,170,157,138,213,101)(14,38,68,169,158,137,214,100)(15,37,69,196,159,136,215,99)(16,36,70,195,160,135,216,98)(17,35,71,194,161,134,217,97)(18,34,72,193,162,133,218,96)(19,33,73,192,163,132,219,95)(20,32,74,191,164,131,220,94)(21,31,75,190,165,130,221,93)(22,30,76,189,166,129,222,92)(23,29,77,188,167,128,223,91)(24,56,78,187,168,127,224,90)(25,55,79,186,141,126,197,89)(26,54,80,185,142,125,198,88)(27,53,81,184,143,124,199,87)(28,52,82,183,144,123,200,86) );
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)], [(1,51,83,182,145,122,201,85),(2,50,84,181,146,121,202,112),(3,49,57,180,147,120,203,111),(4,48,58,179,148,119,204,110),(5,47,59,178,149,118,205,109),(6,46,60,177,150,117,206,108),(7,45,61,176,151,116,207,107),(8,44,62,175,152,115,208,106),(9,43,63,174,153,114,209,105),(10,42,64,173,154,113,210,104),(11,41,65,172,155,140,211,103),(12,40,66,171,156,139,212,102),(13,39,67,170,157,138,213,101),(14,38,68,169,158,137,214,100),(15,37,69,196,159,136,215,99),(16,36,70,195,160,135,216,98),(17,35,71,194,161,134,217,97),(18,34,72,193,162,133,218,96),(19,33,73,192,163,132,219,95),(20,32,74,191,164,131,220,94),(21,31,75,190,165,130,221,93),(22,30,76,189,166,129,222,92),(23,29,77,188,167,128,223,91),(24,56,78,187,168,127,224,90),(25,55,79,186,141,126,197,89),(26,54,80,185,142,125,198,88),(27,53,81,184,143,124,199,87),(28,52,82,183,144,123,200,86)]])
C28⋊C8 is a maximal subgroup of
C4.8Dic28 C56⋊2C8 C56⋊1C8 C4.17D56 C28.53D8 C28.39SD16 C4.Dic28 C28.47D8 C4.D56 C28.2D8 C28.57D8 C28.26Q16 C28.9D8 C28.5Q16 C28.10D8 C8×Dic14 C56⋊11Q8 C8×D28 C8⋊6D28 C56⋊Q8 C8⋊9D28 D7×C4⋊C8 C42.200D14 C42.202D14 C28⋊M4(2) C42.30D14 C42.31D14 C28⋊7M4(2) C42.6Dic7 C42.7Dic7 C42.43D14 C42.187D14 C28.50D8 C28.38SD16 D4.3Dic14 D4×C7⋊C8 C42.47D14 C28⋊3M4(2) C28⋊7D8 D4.1D28 D4.2D28 C28.48SD16 C28.23Q16 Q8.3Dic14 Q8×C7⋊C8 C42.210D14 Q8⋊D28 Q8.1D28 C28⋊7Q16 C42.61D14 D28.23D4 Dic14.4Q8 D28.4Q8 C28⋊2D8 Dic14⋊9D4 C28⋊5SD16 D28⋊5Q8 D28⋊6Q8 C28⋊Q16 Dic14⋊5Q8 Dic14⋊6Q8
C28⋊C8 is a maximal quotient of
C56⋊2C8 C56⋊1C8 C28⋊C16 C56.16Q8 (C2×C28)⋊3C8
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | D7 | M4(2) | Dic7 | D14 | C7⋊C8 | Dic14 | D28 | C4.Dic7 |
| kernel | C28⋊C8 | C2×C7⋊C8 | C4×C28 | C2×C28 | C28 | C28 | C28 | C42 | C14 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 6 | 6 | 12 |
Matrix representation of C28⋊C8 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 8 | 100 |
| 0 | 57 | 77 |
| 69 | 0 | 0 |
| 0 | 30 | 71 |
| 0 | 13 | 83 |
G:=sub<GL(3,GF(113))| [112,0,0,0,8,57,0,100,77],[69,0,0,0,30,13,0,71,83] >;
C28⋊C8 in GAP, Magma, Sage, TeX
C_{28}\rtimes C_8 % in TeX
G:=Group("C28:C8"); // GroupNames label
G:=SmallGroup(224,10);
// by ID
G=gap.SmallGroup(224,10);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,55,86,6917]);
// Polycyclic
G:=Group<a,b|a^28=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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