metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊3M4(2), C42.206D14, C7⋊C8⋊18D4, C7⋊4(C8⋊6D4), (C4×D4).6D7, (D4×C28).7C2, (D4×C14).9C4, C28⋊C8⋊22C2, C4.216(D4×D7), C14.92(C4×D4), C4⋊C4.8Dic7, C2.9(D4×Dic7), C28.375(C2×D4), C4⋊1(C4.Dic7), (C2×D4).6Dic7, C14.40(C8○D4), C28.55D4⋊3C2, (C4×C28).83C22, C22⋊C4.5Dic7, C28.308(C4○D4), C23.9(C2×Dic7), (C2×C28).850C23, (C22×C4).115D14, C14.41(C2×M4(2)), C2.7(Q8.Dic7), C4.135(D4⋊2D7), (C22×C28).100C22, C22.46(C22×Dic7), (C4×C7⋊C8)⋊7C2, (C7×C4⋊C4).12C4, (C7×C22⋊C4).6C4, (C2×C4.Dic7)⋊5C2, (C2×C28).164(C2×C4), C2.9(C2×C4.Dic7), (C2×C7⋊C8).201C22, (C2×C4).20(C2×Dic7), (C22×C14).61(C2×C4), (C2×C4).792(C22×D7), (C2×C14).187(C22×C4), SmallGroup(448,546)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊3M4(2)
G = < a,b,c | a28=b8=c2=1, bab-1=a-1, cac=a15, cbc=b5 >
Subgroups: 340 in 122 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C28, C28, C28, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C7⋊C8, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C8⋊6D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, C4×C7⋊C8, C28⋊C8, C28.55D4, C2×C4.Dic7, D4×C28, C28⋊3M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, M4(2), C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C2×M4(2), C8○D4, C2×Dic7, C22×D7, C8⋊6D4, C4.Dic7, D4×D7, D4⋊2D7, C22×Dic7, C2×C4.Dic7, D4×Dic7, Q8.Dic7, C28⋊3M4(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)
(1 170 167 52 100 78 133 219)(2 169 168 51 101 77 134 218)(3 196 141 50 102 76 135 217)(4 195 142 49 103 75 136 216)(5 194 143 48 104 74 137 215)(6 193 144 47 105 73 138 214)(7 192 145 46 106 72 139 213)(8 191 146 45 107 71 140 212)(9 190 147 44 108 70 113 211)(10 189 148 43 109 69 114 210)(11 188 149 42 110 68 115 209)(12 187 150 41 111 67 116 208)(13 186 151 40 112 66 117 207)(14 185 152 39 85 65 118 206)(15 184 153 38 86 64 119 205)(16 183 154 37 87 63 120 204)(17 182 155 36 88 62 121 203)(18 181 156 35 89 61 122 202)(19 180 157 34 90 60 123 201)(20 179 158 33 91 59 124 200)(21 178 159 32 92 58 125 199)(22 177 160 31 93 57 126 198)(23 176 161 30 94 84 127 197)(24 175 162 29 95 83 128 224)(25 174 163 56 96 82 129 223)(26 173 164 55 97 81 130 222)(27 172 165 54 98 80 131 221)(28 171 166 53 99 79 132 220)
(1 93)(2 108)(3 95)(4 110)(5 97)(6 112)(7 99)(8 86)(9 101)(10 88)(11 103)(12 90)(13 105)(14 92)(15 107)(16 94)(17 109)(18 96)(19 111)(20 98)(21 85)(22 100)(23 87)(24 102)(25 89)(26 104)(27 91)(28 106)(29 50)(30 37)(31 52)(32 39)(33 54)(34 41)(35 56)(36 43)(38 45)(40 47)(42 49)(44 51)(46 53)(48 55)(57 78)(58 65)(59 80)(60 67)(61 82)(62 69)(63 84)(64 71)(66 73)(68 75)(70 77)(72 79)(74 81)(76 83)(113 168)(114 155)(115 142)(116 157)(117 144)(118 159)(119 146)(120 161)(121 148)(122 163)(123 150)(124 165)(125 152)(126 167)(127 154)(128 141)(129 156)(130 143)(131 158)(132 145)(133 160)(134 147)(135 162)(136 149)(137 164)(138 151)(139 166)(140 153)(169 190)(170 177)(171 192)(172 179)(173 194)(174 181)(175 196)(176 183)(178 185)(180 187)(182 189)(184 191)(186 193)(188 195)(197 204)(198 219)(199 206)(200 221)(201 208)(202 223)(203 210)(205 212)(207 214)(209 216)(211 218)(213 220)(215 222)(217 224)
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), (1,170,167,52,100,78,133,219)(2,169,168,51,101,77,134,218)(3,196,141,50,102,76,135,217)(4,195,142,49,103,75,136,216)(5,194,143,48,104,74,137,215)(6,193,144,47,105,73,138,214)(7,192,145,46,106,72,139,213)(8,191,146,45,107,71,140,212)(9,190,147,44,108,70,113,211)(10,189,148,43,109,69,114,210)(11,188,149,42,110,68,115,209)(12,187,150,41,111,67,116,208)(13,186,151,40,112,66,117,207)(14,185,152,39,85,65,118,206)(15,184,153,38,86,64,119,205)(16,183,154,37,87,63,120,204)(17,182,155,36,88,62,121,203)(18,181,156,35,89,61,122,202)(19,180,157,34,90,60,123,201)(20,179,158,33,91,59,124,200)(21,178,159,32,92,58,125,199)(22,177,160,31,93,57,126,198)(23,176,161,30,94,84,127,197)(24,175,162,29,95,83,128,224)(25,174,163,56,96,82,129,223)(26,173,164,55,97,81,130,222)(27,172,165,54,98,80,131,221)(28,171,166,53,99,79,132,220), (1,93)(2,108)(3,95)(4,110)(5,97)(6,112)(7,99)(8,86)(9,101)(10,88)(11,103)(12,90)(13,105)(14,92)(15,107)(16,94)(17,109)(18,96)(19,111)(20,98)(21,85)(22,100)(23,87)(24,102)(25,89)(26,104)(27,91)(28,106)(29,50)(30,37)(31,52)(32,39)(33,54)(34,41)(35,56)(36,43)(38,45)(40,47)(42,49)(44,51)(46,53)(48,55)(57,78)(58,65)(59,80)(60,67)(61,82)(62,69)(63,84)(64,71)(66,73)(68,75)(70,77)(72,79)(74,81)(76,83)(113,168)(114,155)(115,142)(116,157)(117,144)(118,159)(119,146)(120,161)(121,148)(122,163)(123,150)(124,165)(125,152)(126,167)(127,154)(128,141)(129,156)(130,143)(131,158)(132,145)(133,160)(134,147)(135,162)(136,149)(137,164)(138,151)(139,166)(140,153)(169,190)(170,177)(171,192)(172,179)(173,194)(174,181)(175,196)(176,183)(178,185)(180,187)(182,189)(184,191)(186,193)(188,195)(197,204)(198,219)(199,206)(200,221)(201,208)(202,223)(203,210)(205,212)(207,214)(209,216)(211,218)(213,220)(215,222)(217,224)>;
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), (1,170,167,52,100,78,133,219)(2,169,168,51,101,77,134,218)(3,196,141,50,102,76,135,217)(4,195,142,49,103,75,136,216)(5,194,143,48,104,74,137,215)(6,193,144,47,105,73,138,214)(7,192,145,46,106,72,139,213)(8,191,146,45,107,71,140,212)(9,190,147,44,108,70,113,211)(10,189,148,43,109,69,114,210)(11,188,149,42,110,68,115,209)(12,187,150,41,111,67,116,208)(13,186,151,40,112,66,117,207)(14,185,152,39,85,65,118,206)(15,184,153,38,86,64,119,205)(16,183,154,37,87,63,120,204)(17,182,155,36,88,62,121,203)(18,181,156,35,89,61,122,202)(19,180,157,34,90,60,123,201)(20,179,158,33,91,59,124,200)(21,178,159,32,92,58,125,199)(22,177,160,31,93,57,126,198)(23,176,161,30,94,84,127,197)(24,175,162,29,95,83,128,224)(25,174,163,56,96,82,129,223)(26,173,164,55,97,81,130,222)(27,172,165,54,98,80,131,221)(28,171,166,53,99,79,132,220), (1,93)(2,108)(3,95)(4,110)(5,97)(6,112)(7,99)(8,86)(9,101)(10,88)(11,103)(12,90)(13,105)(14,92)(15,107)(16,94)(17,109)(18,96)(19,111)(20,98)(21,85)(22,100)(23,87)(24,102)(25,89)(26,104)(27,91)(28,106)(29,50)(30,37)(31,52)(32,39)(33,54)(34,41)(35,56)(36,43)(38,45)(40,47)(42,49)(44,51)(46,53)(48,55)(57,78)(58,65)(59,80)(60,67)(61,82)(62,69)(63,84)(64,71)(66,73)(68,75)(70,77)(72,79)(74,81)(76,83)(113,168)(114,155)(115,142)(116,157)(117,144)(118,159)(119,146)(120,161)(121,148)(122,163)(123,150)(124,165)(125,152)(126,167)(127,154)(128,141)(129,156)(130,143)(131,158)(132,145)(133,160)(134,147)(135,162)(136,149)(137,164)(138,151)(139,166)(140,153)(169,190)(170,177)(171,192)(172,179)(173,194)(174,181)(175,196)(176,183)(178,185)(180,187)(182,189)(184,191)(186,193)(188,195)(197,204)(198,219)(199,206)(200,221)(201,208)(202,223)(203,210)(205,212)(207,214)(209,216)(211,218)(213,220)(215,222)(217,224) );
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)], [(1,170,167,52,100,78,133,219),(2,169,168,51,101,77,134,218),(3,196,141,50,102,76,135,217),(4,195,142,49,103,75,136,216),(5,194,143,48,104,74,137,215),(6,193,144,47,105,73,138,214),(7,192,145,46,106,72,139,213),(8,191,146,45,107,71,140,212),(9,190,147,44,108,70,113,211),(10,189,148,43,109,69,114,210),(11,188,149,42,110,68,115,209),(12,187,150,41,111,67,116,208),(13,186,151,40,112,66,117,207),(14,185,152,39,85,65,118,206),(15,184,153,38,86,64,119,205),(16,183,154,37,87,63,120,204),(17,182,155,36,88,62,121,203),(18,181,156,35,89,61,122,202),(19,180,157,34,90,60,123,201),(20,179,158,33,91,59,124,200),(21,178,159,32,92,58,125,199),(22,177,160,31,93,57,126,198),(23,176,161,30,94,84,127,197),(24,175,162,29,95,83,128,224),(25,174,163,56,96,82,129,223),(26,173,164,55,97,81,130,222),(27,172,165,54,98,80,131,221),(28,171,166,53,99,79,132,220)], [(1,93),(2,108),(3,95),(4,110),(5,97),(6,112),(7,99),(8,86),(9,101),(10,88),(11,103),(12,90),(13,105),(14,92),(15,107),(16,94),(17,109),(18,96),(19,111),(20,98),(21,85),(22,100),(23,87),(24,102),(25,89),(26,104),(27,91),(28,106),(29,50),(30,37),(31,52),(32,39),(33,54),(34,41),(35,56),(36,43),(38,45),(40,47),(42,49),(44,51),(46,53),(48,55),(57,78),(58,65),(59,80),(60,67),(61,82),(62,69),(63,84),(64,71),(66,73),(68,75),(70,77),(72,79),(74,81),(76,83),(113,168),(114,155),(115,142),(116,157),(117,144),(118,159),(119,146),(120,161),(121,148),(122,163),(123,150),(124,165),(125,152),(126,167),(127,154),(128,141),(129,156),(130,143),(131,158),(132,145),(133,160),(134,147),(135,162),(136,149),(137,164),(138,151),(139,166),(140,153),(169,190),(170,177),(171,192),(172,179),(173,194),(174,181),(175,196),(176,183),(178,185),(180,187),(182,189),(184,191),(186,193),(188,195),(197,204),(198,219),(199,206),(200,221),(201,208),(202,223),(203,210),(205,212),(207,214),(209,216),(211,218),(213,220),(215,222),(217,224)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| 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 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | M4(2) | C4○D4 | D14 | Dic7 | Dic7 | D14 | Dic7 | C8○D4 | C4.Dic7 | D4×D7 | D4⋊2D7 | Q8.Dic7 |
| kernel | C28⋊3M4(2) | C4×C7⋊C8 | C28⋊C8 | C28.55D4 | C2×C4.Dic7 | D4×C28 | C7×C22⋊C4 | C7×C4⋊C4 | D4×C14 | C7⋊C8 | C4×D4 | C28 | C28 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 3 | 6 | 3 | 4 | 24 | 3 | 3 | 6 |
Matrix representation of C28⋊3M4(2) ►in GL4(𝔽113) generated by
| 4 | 0 | 0 | 0 |
| 20 | 85 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 60 | 17 | 0 | 0 |
| 20 | 53 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 95 |
| 1 | 0 | 0 | 0 |
| 86 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [4,20,0,0,0,85,0,0,0,0,0,112,0,0,1,0],[60,20,0,0,17,53,0,0,0,0,18,0,0,0,0,95],[1,86,0,0,0,112,0,0,0,0,0,1,0,0,1,0] >;
C28⋊3M4(2) in GAP, Magma, Sage, TeX
C_{28}\rtimes_3M_4(2) % in TeX
G:=Group("C28:3M4(2)"); // GroupNames label
G:=SmallGroup(448,546);
// by ID
G=gap.SmallGroup(448,546);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,758,219,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^15,c*b*c=b^5>;
// generators/relations