metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊1M4(2), Dic7⋊3M4(2), C42.203D14, C4⋊C8⋊13D7, (C4×D7).8Q8, C4.55(Q8×D7), C4⋊3(C8⋊D7), C28⋊C8⋊14C2, (C4×D7).33D4, C4.207(D4×D7), Dic7⋊C8⋊23C2, D14.1(C4⋊C4), C28.366(C2×D4), (C2×C8).182D14, C28.113(C2×Q8), (C4×Dic7).7C4, (D7×C42).4C2, Dic7.2(C4⋊C4), C7⋊1(C4⋊M4(2)), (C4×C28).63C22, C2.18(D7×M4(2)), C14.8(C2×M4(2)), (C2×C56).213C22, (C2×C28).834C23, (C4×Dic7).277C22, (C7×C4⋊C8)⋊18C2, C2.9(D7×C4⋊C4), (C2×C4×D7).7C4, C14.8(C2×C4⋊C4), (C2×C28).71(C2×C4), (C2×C4).147(C4×D7), (C2×C8⋊D7).8C2, C2.13(C2×C8⋊D7), C22.112(C2×C4×D7), (C2×C7⋊C8).196C22, (C2×C4×D7).280C22, (C2×C14).89(C22×C4), (C2×Dic7).90(C2×C4), (C22×D7).59(C2×C4), (C2×C4).776(C22×D7), SmallGroup(448,371)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊M4(2)
G = < a,b,c | a28=b8=c2=1, bab-1=a-1, cac=a13, cbc=b5 >
Subgroups: 516 in 126 conjugacy classes, 61 normal (37 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, D7, C14, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, Dic7, Dic7, C28, C28, C28, D14, D14, C2×C14, C4⋊C8, C4⋊C8, C2×C42, C2×M4(2), C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×C28, C22×D7, C4⋊M4(2), C8⋊D7, C2×C7⋊C8, C4×Dic7, C4×C28, C2×C56, C2×C4×D7, C28⋊C8, Dic7⋊C8, C7×C4⋊C8, D7×C42, C2×C8⋊D7, C28⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C2×M4(2), C4×D7, C22×D7, C4⋊M4(2), C8⋊D7, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, C2×C8⋊D7, D7×M4(2), C28⋊M4(2)
►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)
(1 172 36 77 136 209 159 104)(2 171 37 76 137 208 160 103)(3 170 38 75 138 207 161 102)(4 169 39 74 139 206 162 101)(5 196 40 73 140 205 163 100)(6 195 41 72 113 204 164 99)(7 194 42 71 114 203 165 98)(8 193 43 70 115 202 166 97)(9 192 44 69 116 201 167 96)(10 191 45 68 117 200 168 95)(11 190 46 67 118 199 141 94)(12 189 47 66 119 198 142 93)(13 188 48 65 120 197 143 92)(14 187 49 64 121 224 144 91)(15 186 50 63 122 223 145 90)(16 185 51 62 123 222 146 89)(17 184 52 61 124 221 147 88)(18 183 53 60 125 220 148 87)(19 182 54 59 126 219 149 86)(20 181 55 58 127 218 150 85)(21 180 56 57 128 217 151 112)(22 179 29 84 129 216 152 111)(23 178 30 83 130 215 153 110)(24 177 31 82 131 214 154 109)(25 176 32 81 132 213 155 108)(26 175 33 80 133 212 156 107)(27 174 34 79 134 211 157 106)(28 173 35 78 135 210 158 105)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 43)(30 56)(31 41)(32 54)(33 39)(34 52)(35 37)(36 50)(38 48)(40 46)(42 44)(45 55)(47 53)(49 51)(57 110)(58 95)(59 108)(60 93)(61 106)(62 91)(63 104)(64 89)(65 102)(66 87)(67 100)(68 85)(69 98)(70 111)(71 96)(72 109)(73 94)(74 107)(75 92)(76 105)(77 90)(78 103)(79 88)(80 101)(81 86)(82 99)(83 112)(84 97)(113 131)(114 116)(115 129)(117 127)(118 140)(119 125)(120 138)(121 123)(122 136)(124 134)(126 132)(128 130)(133 139)(135 137)(141 163)(142 148)(143 161)(144 146)(145 159)(147 157)(149 155)(150 168)(151 153)(152 166)(154 164)(156 162)(158 160)(165 167)(169 212)(170 197)(171 210)(172 223)(173 208)(174 221)(175 206)(176 219)(177 204)(178 217)(179 202)(180 215)(181 200)(182 213)(183 198)(184 211)(185 224)(186 209)(187 222)(188 207)(189 220)(190 205)(191 218)(192 203)(193 216)(194 201)(195 214)(196 199)
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), (1,172,36,77,136,209,159,104)(2,171,37,76,137,208,160,103)(3,170,38,75,138,207,161,102)(4,169,39,74,139,206,162,101)(5,196,40,73,140,205,163,100)(6,195,41,72,113,204,164,99)(7,194,42,71,114,203,165,98)(8,193,43,70,115,202,166,97)(9,192,44,69,116,201,167,96)(10,191,45,68,117,200,168,95)(11,190,46,67,118,199,141,94)(12,189,47,66,119,198,142,93)(13,188,48,65,120,197,143,92)(14,187,49,64,121,224,144,91)(15,186,50,63,122,223,145,90)(16,185,51,62,123,222,146,89)(17,184,52,61,124,221,147,88)(18,183,53,60,125,220,148,87)(19,182,54,59,126,219,149,86)(20,181,55,58,127,218,150,85)(21,180,56,57,128,217,151,112)(22,179,29,84,129,216,152,111)(23,178,30,83,130,215,153,110)(24,177,31,82,131,214,154,109)(25,176,32,81,132,213,155,108)(26,175,33,80,133,212,156,107)(27,174,34,79,134,211,157,106)(28,173,35,78,135,210,158,105), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,43)(30,56)(31,41)(32,54)(33,39)(34,52)(35,37)(36,50)(38,48)(40,46)(42,44)(45,55)(47,53)(49,51)(57,110)(58,95)(59,108)(60,93)(61,106)(62,91)(63,104)(64,89)(65,102)(66,87)(67,100)(68,85)(69,98)(70,111)(71,96)(72,109)(73,94)(74,107)(75,92)(76,105)(77,90)(78,103)(79,88)(80,101)(81,86)(82,99)(83,112)(84,97)(113,131)(114,116)(115,129)(117,127)(118,140)(119,125)(120,138)(121,123)(122,136)(124,134)(126,132)(128,130)(133,139)(135,137)(141,163)(142,148)(143,161)(144,146)(145,159)(147,157)(149,155)(150,168)(151,153)(152,166)(154,164)(156,162)(158,160)(165,167)(169,212)(170,197)(171,210)(172,223)(173,208)(174,221)(175,206)(176,219)(177,204)(178,217)(179,202)(180,215)(181,200)(182,213)(183,198)(184,211)(185,224)(186,209)(187,222)(188,207)(189,220)(190,205)(191,218)(192,203)(193,216)(194,201)(195,214)(196,199)>;
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), (1,172,36,77,136,209,159,104)(2,171,37,76,137,208,160,103)(3,170,38,75,138,207,161,102)(4,169,39,74,139,206,162,101)(5,196,40,73,140,205,163,100)(6,195,41,72,113,204,164,99)(7,194,42,71,114,203,165,98)(8,193,43,70,115,202,166,97)(9,192,44,69,116,201,167,96)(10,191,45,68,117,200,168,95)(11,190,46,67,118,199,141,94)(12,189,47,66,119,198,142,93)(13,188,48,65,120,197,143,92)(14,187,49,64,121,224,144,91)(15,186,50,63,122,223,145,90)(16,185,51,62,123,222,146,89)(17,184,52,61,124,221,147,88)(18,183,53,60,125,220,148,87)(19,182,54,59,126,219,149,86)(20,181,55,58,127,218,150,85)(21,180,56,57,128,217,151,112)(22,179,29,84,129,216,152,111)(23,178,30,83,130,215,153,110)(24,177,31,82,131,214,154,109)(25,176,32,81,132,213,155,108)(26,175,33,80,133,212,156,107)(27,174,34,79,134,211,157,106)(28,173,35,78,135,210,158,105), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,43)(30,56)(31,41)(32,54)(33,39)(34,52)(35,37)(36,50)(38,48)(40,46)(42,44)(45,55)(47,53)(49,51)(57,110)(58,95)(59,108)(60,93)(61,106)(62,91)(63,104)(64,89)(65,102)(66,87)(67,100)(68,85)(69,98)(70,111)(71,96)(72,109)(73,94)(74,107)(75,92)(76,105)(77,90)(78,103)(79,88)(80,101)(81,86)(82,99)(83,112)(84,97)(113,131)(114,116)(115,129)(117,127)(118,140)(119,125)(120,138)(121,123)(122,136)(124,134)(126,132)(128,130)(133,139)(135,137)(141,163)(142,148)(143,161)(144,146)(145,159)(147,157)(149,155)(150,168)(151,153)(152,166)(154,164)(156,162)(158,160)(165,167)(169,212)(170,197)(171,210)(172,223)(173,208)(174,221)(175,206)(176,219)(177,204)(178,217)(179,202)(180,215)(181,200)(182,213)(183,198)(184,211)(185,224)(186,209)(187,222)(188,207)(189,220)(190,205)(191,218)(192,203)(193,216)(194,201)(195,214)(196,199) );
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)], [(1,172,36,77,136,209,159,104),(2,171,37,76,137,208,160,103),(3,170,38,75,138,207,161,102),(4,169,39,74,139,206,162,101),(5,196,40,73,140,205,163,100),(6,195,41,72,113,204,164,99),(7,194,42,71,114,203,165,98),(8,193,43,70,115,202,166,97),(9,192,44,69,116,201,167,96),(10,191,45,68,117,200,168,95),(11,190,46,67,118,199,141,94),(12,189,47,66,119,198,142,93),(13,188,48,65,120,197,143,92),(14,187,49,64,121,224,144,91),(15,186,50,63,122,223,145,90),(16,185,51,62,123,222,146,89),(17,184,52,61,124,221,147,88),(18,183,53,60,125,220,148,87),(19,182,54,59,126,219,149,86),(20,181,55,58,127,218,150,85),(21,180,56,57,128,217,151,112),(22,179,29,84,129,216,152,111),(23,178,30,83,130,215,153,110),(24,177,31,82,131,214,154,109),(25,176,32,81,132,213,155,108),(26,175,33,80,133,212,156,107),(27,174,34,79,134,211,157,106),(28,173,35,78,135,210,158,105)], [(1,15),(2,28),(3,13),(4,26),(5,11),(6,24),(7,9),(8,22),(10,20),(12,18),(14,16),(17,27),(19,25),(21,23),(29,43),(30,56),(31,41),(32,54),(33,39),(34,52),(35,37),(36,50),(38,48),(40,46),(42,44),(45,55),(47,53),(49,51),(57,110),(58,95),(59,108),(60,93),(61,106),(62,91),(63,104),(64,89),(65,102),(66,87),(67,100),(68,85),(69,98),(70,111),(71,96),(72,109),(73,94),(74,107),(75,92),(76,105),(77,90),(78,103),(79,88),(80,101),(81,86),(82,99),(83,112),(84,97),(113,131),(114,116),(115,129),(117,127),(118,140),(119,125),(120,138),(121,123),(122,136),(124,134),(126,132),(128,130),(133,139),(135,137),(141,163),(142,148),(143,161),(144,146),(145,159),(147,157),(149,155),(150,168),(151,153),(152,166),(154,164),(156,162),(158,160),(165,167),(169,212),(170,197),(171,210),(172,223),(173,208),(174,221),(175,206),(176,219),(177,204),(178,217),(179,202),(180,215),(181,200),(182,213),(183,198),(184,211),(185,224),(186,209),(187,222),(188,207),(189,220),(190,205),(191,218),(192,203),(193,216),(194,201),(195,214),(196,199)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D7 | M4(2) | M4(2) | D14 | D14 | C4×D7 | C8⋊D7 | D4×D7 | Q8×D7 | D7×M4(2) |
| kernel | C28⋊M4(2) | C28⋊C8 | Dic7⋊C8 | C7×C4⋊C8 | D7×C42 | C2×C8⋊D7 | C4×Dic7 | C2×C4×D7 | C4×D7 | C4×D7 | C4⋊C8 | Dic7 | C28 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 3 | 4 | 4 | 3 | 6 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of C28⋊M4(2) ►in GL4(𝔽113) generated by
| 16 | 12 | 0 | 0 |
| 101 | 97 | 0 | 0 |
| 0 | 0 | 98 | 24 |
| 0 | 0 | 65 | 24 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 58 | 5 |
| 0 | 0 | 76 | 55 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 23 | 10 |
| 0 | 0 | 15 | 90 |
G:=sub<GL(4,GF(113))| [16,101,0,0,12,97,0,0,0,0,98,65,0,0,24,24],[0,112,0,0,112,0,0,0,0,0,58,76,0,0,5,55],[112,0,0,0,0,112,0,0,0,0,23,15,0,0,10,90] >;
C28⋊M4(2) in GAP, Magma, Sage, TeX
C_{28}\rtimes M_4(2) % in TeX
G:=Group("C28:M4(2)"); // GroupNames label
G:=SmallGroup(448,371);
// by ID
G=gap.SmallGroup(448,371);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^5>;
// generators/relations