direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C28, C42.9C28, C28.30C42, C8⋊6(C2×C28), (C4×C56)⋊29C2, C56⋊28(C2×C4), (C4×C8)⋊13C14, C4.4(C4×C28), (C4×C28).24C4, C8⋊C4⋊12C14, C22.4(C4×C28), C42.58(C2×C14), (C22×C28).18C4, (C2×C42).12C14, C23.27(C2×C28), C14.34(C2×C42), (C2×C14).11C42, C4.33(C22×C28), (C22×C4).10C28, C2.2(C14×M4(2)), (C4×C28).299C22, (C2×C28).979C23, C28.191(C22×C4), (C2×C56).445C22, C14.46(C2×M4(2)), (C2×M4(2)).16C14, (C14×M4(2)).35C2, C22.17(C22×C28), (C22×C28).578C22, C2.6(C2×C4×C28), (C2×C4×C28).35C2, (C7×C8⋊C4)⋊26C2, (C2×C8).99(C2×C14), (C2×C4).45(C2×C28), (C2×C28).347(C2×C4), (C22×C4).108(C2×C14), (C2×C4).147(C22×C14), (C2×C14).229(C22×C4), (C22×C14).113(C2×C4), SmallGroup(448,812)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C28
G = < a,b,c | a28=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Subgroups: 162 in 142 conjugacy classes, 122 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C56, C2×C28, C2×C28, C2×C28, C22×C14, C4×M4(2), C4×C28, C4×C28, C2×C56, C7×M4(2), C22×C28, C22×C28, C4×C56, C7×C8⋊C4, C2×C4×C28, C14×M4(2), M4(2)×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C42, M4(2), C22×C4, C28, C2×C14, C2×C42, C2×M4(2), C2×C28, C22×C14, C4×M4(2), C4×C28, C7×M4(2), C22×C28, C2×C4×C28, C14×M4(2), M4(2)×C28
►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)
(1 109 219 134 148 74 48 182)(2 110 220 135 149 75 49 183)(3 111 221 136 150 76 50 184)(4 112 222 137 151 77 51 185)(5 85 223 138 152 78 52 186)(6 86 224 139 153 79 53 187)(7 87 197 140 154 80 54 188)(8 88 198 113 155 81 55 189)(9 89 199 114 156 82 56 190)(10 90 200 115 157 83 29 191)(11 91 201 116 158 84 30 192)(12 92 202 117 159 57 31 193)(13 93 203 118 160 58 32 194)(14 94 204 119 161 59 33 195)(15 95 205 120 162 60 34 196)(16 96 206 121 163 61 35 169)(17 97 207 122 164 62 36 170)(18 98 208 123 165 63 37 171)(19 99 209 124 166 64 38 172)(20 100 210 125 167 65 39 173)(21 101 211 126 168 66 40 174)(22 102 212 127 141 67 41 175)(23 103 213 128 142 68 42 176)(24 104 214 129 143 69 43 177)(25 105 215 130 144 70 44 178)(26 106 216 131 145 71 45 179)(27 107 217 132 146 72 46 180)(28 108 218 133 147 73 47 181)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(113 175)(114 176)(115 177)(116 178)(117 179)(118 180)(119 181)(120 182)(121 183)(122 184)(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 169)(136 170)(137 171)(138 172)(139 173)(140 174)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
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), (1,109,219,134,148,74,48,182)(2,110,220,135,149,75,49,183)(3,111,221,136,150,76,50,184)(4,112,222,137,151,77,51,185)(5,85,223,138,152,78,52,186)(6,86,224,139,153,79,53,187)(7,87,197,140,154,80,54,188)(8,88,198,113,155,81,55,189)(9,89,199,114,156,82,56,190)(10,90,200,115,157,83,29,191)(11,91,201,116,158,84,30,192)(12,92,202,117,159,57,31,193)(13,93,203,118,160,58,32,194)(14,94,204,119,161,59,33,195)(15,95,205,120,162,60,34,196)(16,96,206,121,163,61,35,169)(17,97,207,122,164,62,36,170)(18,98,208,123,165,63,37,171)(19,99,209,124,166,64,38,172)(20,100,210,125,167,65,39,173)(21,101,211,126,168,66,40,174)(22,102,212,127,141,67,41,175)(23,103,213,128,142,68,42,176)(24,104,214,129,143,69,43,177)(25,105,215,130,144,70,44,178)(26,106,216,131,145,71,45,179)(27,107,217,132,146,72,46,180)(28,108,218,133,147,73,47,181), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,169)(136,170)(137,171)(138,172)(139,173)(140,174)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224)>;
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), (1,109,219,134,148,74,48,182)(2,110,220,135,149,75,49,183)(3,111,221,136,150,76,50,184)(4,112,222,137,151,77,51,185)(5,85,223,138,152,78,52,186)(6,86,224,139,153,79,53,187)(7,87,197,140,154,80,54,188)(8,88,198,113,155,81,55,189)(9,89,199,114,156,82,56,190)(10,90,200,115,157,83,29,191)(11,91,201,116,158,84,30,192)(12,92,202,117,159,57,31,193)(13,93,203,118,160,58,32,194)(14,94,204,119,161,59,33,195)(15,95,205,120,162,60,34,196)(16,96,206,121,163,61,35,169)(17,97,207,122,164,62,36,170)(18,98,208,123,165,63,37,171)(19,99,209,124,166,64,38,172)(20,100,210,125,167,65,39,173)(21,101,211,126,168,66,40,174)(22,102,212,127,141,67,41,175)(23,103,213,128,142,68,42,176)(24,104,214,129,143,69,43,177)(25,105,215,130,144,70,44,178)(26,106,216,131,145,71,45,179)(27,107,217,132,146,72,46,180)(28,108,218,133,147,73,47,181), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,169)(136,170)(137,171)(138,172)(139,173)(140,174)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224) );
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)], [(1,109,219,134,148,74,48,182),(2,110,220,135,149,75,49,183),(3,111,221,136,150,76,50,184),(4,112,222,137,151,77,51,185),(5,85,223,138,152,78,52,186),(6,86,224,139,153,79,53,187),(7,87,197,140,154,80,54,188),(8,88,198,113,155,81,55,189),(9,89,199,114,156,82,56,190),(10,90,200,115,157,83,29,191),(11,91,201,116,158,84,30,192),(12,92,202,117,159,57,31,193),(13,93,203,118,160,58,32,194),(14,94,204,119,161,59,33,195),(15,95,205,120,162,60,34,196),(16,96,206,121,163,61,35,169),(17,97,207,122,164,62,36,170),(18,98,208,123,165,63,37,171),(19,99,209,124,166,64,38,172),(20,100,210,125,167,65,39,173),(21,101,211,126,168,66,40,174),(22,102,212,127,141,67,41,175),(23,103,213,128,142,68,42,176),(24,104,214,129,143,69,43,177),(25,105,215,130,144,70,44,178),(26,106,216,131,145,71,45,179),(27,107,217,132,146,72,46,180),(28,108,218,133,147,73,47,181)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(113,175),(114,176),(115,177),(116,178),(117,179),(118,180),(119,181),(120,182),(121,183),(122,184),(123,185),(124,186),(125,187),(126,188),(127,189),(128,190),(129,191),(130,192),(131,193),(132,194),(133,195),(134,196),(135,169),(136,170),(137,171),(138,172),(139,173),(140,174),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)]])
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 7A | ··· | 7F | 8A | ··· | 8P | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28BT | 28BU | ··· | 28DD | 56A | ··· | 56CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | M4(2) | C7×M4(2) |
| kernel | M4(2)×C28 | C4×C56 | C7×C8⋊C4 | C2×C4×C28 | C14×M4(2) | C4×C28 | C7×M4(2) | C22×C28 | C4×M4(2) | C4×C8 | C8⋊C4 | C2×C42 | C2×M4(2) | C42 | M4(2) | C22×C4 | C28 | C4 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 16 | 4 | 6 | 12 | 12 | 6 | 12 | 24 | 96 | 24 | 8 | 48 |
Matrix representation of M4(2)×C28 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 112 | 0 | 0 |
| 0 | 0 | 111 |
| 0 | 64 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 112 |
G:=sub<GL(3,GF(113))| [98,0,0,0,28,0,0,0,28],[112,0,0,0,0,64,0,111,0],[1,0,0,0,1,0,0,0,112] >;
M4(2)×C28 in GAP, Magma, Sage, TeX
M_4(2)\times C_{28} % in TeX
G:=Group("M4(2)xC28"); // GroupNames label
G:=SmallGroup(448,812);
// by ID
G=gap.SmallGroup(448,812);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,792,4790,172]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations