metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊2M4(2), C42.204D14, C7⋊C8⋊17D4, C4⋊C8⋊14D7, C7⋊3(C8⋊6D4), D14⋊C8⋊24C2, C4⋊1(C8⋊D7), (C4×D28).9C2, C14.48(C4×D4), C4.208(D4×D7), D14⋊C4.12C4, (C2×D28).13C4, (C2×C8).183D14, C28.367(C2×D4), C4⋊Dic7.19C4, C14.29(C8○D4), (C4×C28).64C22, C14.9(C2×M4(2)), C28.336(C4○D4), C2.8(D28⋊C4), (C2×C56).214C22, (C2×C28).835C23, C2.13(D28.C4), C4.56(Q8⋊2D7), (C4×C7⋊C8)⋊5C2, (C7×C4⋊C8)⋊19C2, (C2×C4).73(C4×D7), (C2×C8⋊D7)⋊22C2, C2.14(C2×C8⋊D7), C22.113(C2×C4×D7), (C2×C28).159(C2×C4), (C2×C7⋊C8).307C22, (C2×C4×D7).181C22, (C2×C14).90(C22×C4), (C2×Dic7).23(C2×C4), (C22×D7).16(C2×C4), (C2×C4).777(C22×D7), SmallGroup(448,372)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊2M4(2)
G = < a,b,c | a28=b8=c2=1, bab-1=a13, cac=a-1, cbc=b5 >
Subgroups: 580 in 122 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, Dic7, C28, C28, C28, D14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C7⋊C8, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, C8⋊6D4, C8⋊D7, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C4×C7⋊C8, D14⋊C8, C7×C4⋊C8, C4×D28, C2×C8⋊D7, C28⋊2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, M4(2), C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×M4(2), C8○D4, C4×D7, C22×D7, C8⋊6D4, C8⋊D7, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4, C2×C8⋊D7, D28.C4, C28⋊2M4(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 58 98 208 153 130 51 194)(2 71 99 221 154 115 52 179)(3 84 100 206 155 128 53 192)(4 69 101 219 156 113 54 177)(5 82 102 204 157 126 55 190)(6 67 103 217 158 139 56 175)(7 80 104 202 159 124 29 188)(8 65 105 215 160 137 30 173)(9 78 106 200 161 122 31 186)(10 63 107 213 162 135 32 171)(11 76 108 198 163 120 33 184)(12 61 109 211 164 133 34 169)(13 74 110 224 165 118 35 182)(14 59 111 209 166 131 36 195)(15 72 112 222 167 116 37 180)(16 57 85 207 168 129 38 193)(17 70 86 220 141 114 39 178)(18 83 87 205 142 127 40 191)(19 68 88 218 143 140 41 176)(20 81 89 203 144 125 42 189)(21 66 90 216 145 138 43 174)(22 79 91 201 146 123 44 187)(23 64 92 214 147 136 45 172)(24 77 93 199 148 121 46 185)(25 62 94 212 149 134 47 170)(26 75 95 197 150 119 48 183)(27 60 96 210 151 132 49 196)(28 73 97 223 152 117 50 181)
(1 8)(2 7)(3 6)(4 5)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(53 56)(54 55)(57 138)(58 137)(59 136)(60 135)(61 134)(62 133)(63 132)(64 131)(65 130)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 120)(76 119)(77 118)(78 117)(79 116)(80 115)(81 114)(82 113)(83 140)(84 139)(85 90)(86 89)(87 88)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(141 144)(142 143)(145 168)(146 167)(147 166)(148 165)(149 164)(150 163)(151 162)(152 161)(153 160)(154 159)(155 158)(156 157)(169 212)(170 211)(171 210)(172 209)(173 208)(174 207)(175 206)(176 205)(177 204)(178 203)(179 202)(180 201)(181 200)(182 199)(183 198)(184 197)(185 224)(186 223)(187 222)(188 221)(189 220)(190 219)(191 218)(192 217)(193 216)(194 215)(195 214)(196 213)
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), (1,58,98,208,153,130,51,194)(2,71,99,221,154,115,52,179)(3,84,100,206,155,128,53,192)(4,69,101,219,156,113,54,177)(5,82,102,204,157,126,55,190)(6,67,103,217,158,139,56,175)(7,80,104,202,159,124,29,188)(8,65,105,215,160,137,30,173)(9,78,106,200,161,122,31,186)(10,63,107,213,162,135,32,171)(11,76,108,198,163,120,33,184)(12,61,109,211,164,133,34,169)(13,74,110,224,165,118,35,182)(14,59,111,209,166,131,36,195)(15,72,112,222,167,116,37,180)(16,57,85,207,168,129,38,193)(17,70,86,220,141,114,39,178)(18,83,87,205,142,127,40,191)(19,68,88,218,143,140,41,176)(20,81,89,203,144,125,42,189)(21,66,90,216,145,138,43,174)(22,79,91,201,146,123,44,187)(23,64,92,214,147,136,45,172)(24,77,93,199,148,121,46,185)(25,62,94,212,149,134,47,170)(26,75,95,197,150,119,48,183)(27,60,96,210,151,132,49,196)(28,73,97,223,152,117,50,181), (1,8)(2,7)(3,6)(4,5)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(53,56)(54,55)(57,138)(58,137)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,140)(84,139)(85,90)(86,89)(87,88)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(141,144)(142,143)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(169,212)(170,211)(171,210)(172,209)(173,208)(174,207)(175,206)(176,205)(177,204)(178,203)(179,202)(180,201)(181,200)(182,199)(183,198)(184,197)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)(194,215)(195,214)(196,213)>;
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), (1,58,98,208,153,130,51,194)(2,71,99,221,154,115,52,179)(3,84,100,206,155,128,53,192)(4,69,101,219,156,113,54,177)(5,82,102,204,157,126,55,190)(6,67,103,217,158,139,56,175)(7,80,104,202,159,124,29,188)(8,65,105,215,160,137,30,173)(9,78,106,200,161,122,31,186)(10,63,107,213,162,135,32,171)(11,76,108,198,163,120,33,184)(12,61,109,211,164,133,34,169)(13,74,110,224,165,118,35,182)(14,59,111,209,166,131,36,195)(15,72,112,222,167,116,37,180)(16,57,85,207,168,129,38,193)(17,70,86,220,141,114,39,178)(18,83,87,205,142,127,40,191)(19,68,88,218,143,140,41,176)(20,81,89,203,144,125,42,189)(21,66,90,216,145,138,43,174)(22,79,91,201,146,123,44,187)(23,64,92,214,147,136,45,172)(24,77,93,199,148,121,46,185)(25,62,94,212,149,134,47,170)(26,75,95,197,150,119,48,183)(27,60,96,210,151,132,49,196)(28,73,97,223,152,117,50,181), (1,8)(2,7)(3,6)(4,5)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(53,56)(54,55)(57,138)(58,137)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,140)(84,139)(85,90)(86,89)(87,88)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(141,144)(142,143)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(169,212)(170,211)(171,210)(172,209)(173,208)(174,207)(175,206)(176,205)(177,204)(178,203)(179,202)(180,201)(181,200)(182,199)(183,198)(184,197)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)(194,215)(195,214)(196,213) );
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)], [(1,58,98,208,153,130,51,194),(2,71,99,221,154,115,52,179),(3,84,100,206,155,128,53,192),(4,69,101,219,156,113,54,177),(5,82,102,204,157,126,55,190),(6,67,103,217,158,139,56,175),(7,80,104,202,159,124,29,188),(8,65,105,215,160,137,30,173),(9,78,106,200,161,122,31,186),(10,63,107,213,162,135,32,171),(11,76,108,198,163,120,33,184),(12,61,109,211,164,133,34,169),(13,74,110,224,165,118,35,182),(14,59,111,209,166,131,36,195),(15,72,112,222,167,116,37,180),(16,57,85,207,168,129,38,193),(17,70,86,220,141,114,39,178),(18,83,87,205,142,127,40,191),(19,68,88,218,143,140,41,176),(20,81,89,203,144,125,42,189),(21,66,90,216,145,138,43,174),(22,79,91,201,146,123,44,187),(23,64,92,214,147,136,45,172),(24,77,93,199,148,121,46,185),(25,62,94,212,149,134,47,170),(26,75,95,197,150,119,48,183),(27,60,96,210,151,132,49,196),(28,73,97,223,152,117,50,181)], [(1,8),(2,7),(3,6),(4,5),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(53,56),(54,55),(57,138),(58,137),(59,136),(60,135),(61,134),(62,133),(63,132),(64,131),(65,130),(66,129),(67,128),(68,127),(69,126),(70,125),(71,124),(72,123),(73,122),(74,121),(75,120),(76,119),(77,118),(78,117),(79,116),(80,115),(81,114),(82,113),(83,140),(84,139),(85,90),(86,89),(87,88),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102),(141,144),(142,143),(145,168),(146,167),(147,166),(148,165),(149,164),(150,163),(151,162),(152,161),(153,160),(154,159),(155,158),(156,157),(169,212),(170,211),(171,210),(172,209),(173,208),(174,207),(175,206),(176,205),(177,204),(178,203),(179,202),(180,201),(181,200),(182,199),(183,198),(184,197),(185,224),(186,223),(187,222),(188,221),(189,220),(190,219),(191,218),(192,217),(193,216),(194,215),(195,214),(196,213)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 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 | C4 | D4 | D7 | M4(2) | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | C8⋊D7 | D4×D7 | Q8⋊2D7 | D28.C4 |
| kernel | C28⋊2M4(2) | C4×C7⋊C8 | D14⋊C8 | C7×C4⋊C8 | C4×D28 | C2×C8⋊D7 | C4⋊Dic7 | D14⋊C4 | C2×D28 | C7⋊C8 | C4⋊C8 | C28 | C28 | C42 | C2×C8 | C14 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of C28⋊2M4(2) ►in GL4(𝔽113) generated by
| 112 | 24 | 0 | 0 |
| 89 | 10 | 0 | 0 |
| 0 | 0 | 66 | 4 |
| 0 | 0 | 69 | 47 |
| 33 | 82 | 0 | 0 |
| 83 | 80 | 0 | 0 |
| 0 | 0 | 58 | 72 |
| 0 | 0 | 112 | 55 |
| 1 | 0 | 0 | 0 |
| 24 | 112 | 0 | 0 |
| 0 | 0 | 47 | 32 |
| 0 | 0 | 44 | 66 |
G:=sub<GL(4,GF(113))| [112,89,0,0,24,10,0,0,0,0,66,69,0,0,4,47],[33,83,0,0,82,80,0,0,0,0,58,112,0,0,72,55],[1,24,0,0,0,112,0,0,0,0,47,44,0,0,32,66] >;
C28⋊2M4(2) in GAP, Magma, Sage, TeX
C_{28}\rtimes_2M_4(2) % in TeX
G:=Group("C28:2M4(2)"); // GroupNames label
G:=SmallGroup(448,372);
// by ID
G=gap.SmallGroup(448,372);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,758,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations