metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.44D4, C14.1Q16, Dic14⋊2C4, C2.1Dic28, C14.1SD16, C22.7D28, C4.7(C4×D7), (C2×C8).2D7, (C2×C56).2C2, C28.17(C2×C4), C7⋊2(Q8⋊C4), (C2×C4).67D14, (C2×C14).12D4, C4⋊Dic7.1C2, C2.7(D14⋊C4), C2.1(C56⋊C2), C4.19(C7⋊D4), C14.5(C22⋊C4), (C2×C28).80C22, (C2×Dic14).1C2, SmallGroup(224,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.44D4
G = < a,b,c | a28=b4=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a21b-1 >
Smallest permutation representation of C28.44D4
►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)
(1 161 74 220)(2 160 75 219)(3 159 76 218)(4 158 77 217)(5 157 78 216)(6 156 79 215)(7 155 80 214)(8 154 81 213)(9 153 82 212)(10 152 83 211)(11 151 84 210)(12 150 57 209)(13 149 58 208)(14 148 59 207)(15 147 60 206)(16 146 61 205)(17 145 62 204)(18 144 63 203)(19 143 64 202)(20 142 65 201)(21 141 66 200)(22 168 67 199)(23 167 68 198)(24 166 69 197)(25 165 70 224)(26 164 71 223)(27 163 72 222)(28 162 73 221)(29 179 118 105)(30 178 119 104)(31 177 120 103)(32 176 121 102)(33 175 122 101)(34 174 123 100)(35 173 124 99)(36 172 125 98)(37 171 126 97)(38 170 127 96)(39 169 128 95)(40 196 129 94)(41 195 130 93)(42 194 131 92)(43 193 132 91)(44 192 133 90)(45 191 134 89)(46 190 135 88)(47 189 136 87)(48 188 137 86)(49 187 138 85)(50 186 139 112)(51 185 140 111)(52 184 113 110)(53 183 114 109)(54 182 115 108)(55 181 116 107)(56 180 117 106)
(1 42 15 56)(2 41 16 55)(3 40 17 54)(4 39 18 53)(5 38 19 52)(6 37 20 51)(7 36 21 50)(8 35 22 49)(9 34 23 48)(10 33 24 47)(11 32 25 46)(12 31 26 45)(13 30 27 44)(14 29 28 43)(57 120 71 134)(58 119 72 133)(59 118 73 132)(60 117 74 131)(61 116 75 130)(62 115 76 129)(63 114 77 128)(64 113 78 127)(65 140 79 126)(66 139 80 125)(67 138 81 124)(68 137 82 123)(69 136 83 122)(70 135 84 121)(85 161 99 147)(86 160 100 146)(87 159 101 145)(88 158 102 144)(89 157 103 143)(90 156 104 142)(91 155 105 141)(92 154 106 168)(93 153 107 167)(94 152 108 166)(95 151 109 165)(96 150 110 164)(97 149 111 163)(98 148 112 162)(169 210 183 224)(170 209 184 223)(171 208 185 222)(172 207 186 221)(173 206 187 220)(174 205 188 219)(175 204 189 218)(176 203 190 217)(177 202 191 216)(178 201 192 215)(179 200 193 214)(180 199 194 213)(181 198 195 212)(182 197 196 211)
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), (1,161,74,220)(2,160,75,219)(3,159,76,218)(4,158,77,217)(5,157,78,216)(6,156,79,215)(7,155,80,214)(8,154,81,213)(9,153,82,212)(10,152,83,211)(11,151,84,210)(12,150,57,209)(13,149,58,208)(14,148,59,207)(15,147,60,206)(16,146,61,205)(17,145,62,204)(18,144,63,203)(19,143,64,202)(20,142,65,201)(21,141,66,200)(22,168,67,199)(23,167,68,198)(24,166,69,197)(25,165,70,224)(26,164,71,223)(27,163,72,222)(28,162,73,221)(29,179,118,105)(30,178,119,104)(31,177,120,103)(32,176,121,102)(33,175,122,101)(34,174,123,100)(35,173,124,99)(36,172,125,98)(37,171,126,97)(38,170,127,96)(39,169,128,95)(40,196,129,94)(41,195,130,93)(42,194,131,92)(43,193,132,91)(44,192,133,90)(45,191,134,89)(46,190,135,88)(47,189,136,87)(48,188,137,86)(49,187,138,85)(50,186,139,112)(51,185,140,111)(52,184,113,110)(53,183,114,109)(54,182,115,108)(55,181,116,107)(56,180,117,106), (1,42,15,56)(2,41,16,55)(3,40,17,54)(4,39,18,53)(5,38,19,52)(6,37,20,51)(7,36,21,50)(8,35,22,49)(9,34,23,48)(10,33,24,47)(11,32,25,46)(12,31,26,45)(13,30,27,44)(14,29,28,43)(57,120,71,134)(58,119,72,133)(59,118,73,132)(60,117,74,131)(61,116,75,130)(62,115,76,129)(63,114,77,128)(64,113,78,127)(65,140,79,126)(66,139,80,125)(67,138,81,124)(68,137,82,123)(69,136,83,122)(70,135,84,121)(85,161,99,147)(86,160,100,146)(87,159,101,145)(88,158,102,144)(89,157,103,143)(90,156,104,142)(91,155,105,141)(92,154,106,168)(93,153,107,167)(94,152,108,166)(95,151,109,165)(96,150,110,164)(97,149,111,163)(98,148,112,162)(169,210,183,224)(170,209,184,223)(171,208,185,222)(172,207,186,221)(173,206,187,220)(174,205,188,219)(175,204,189,218)(176,203,190,217)(177,202,191,216)(178,201,192,215)(179,200,193,214)(180,199,194,213)(181,198,195,212)(182,197,196,211)>;
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), (1,161,74,220)(2,160,75,219)(3,159,76,218)(4,158,77,217)(5,157,78,216)(6,156,79,215)(7,155,80,214)(8,154,81,213)(9,153,82,212)(10,152,83,211)(11,151,84,210)(12,150,57,209)(13,149,58,208)(14,148,59,207)(15,147,60,206)(16,146,61,205)(17,145,62,204)(18,144,63,203)(19,143,64,202)(20,142,65,201)(21,141,66,200)(22,168,67,199)(23,167,68,198)(24,166,69,197)(25,165,70,224)(26,164,71,223)(27,163,72,222)(28,162,73,221)(29,179,118,105)(30,178,119,104)(31,177,120,103)(32,176,121,102)(33,175,122,101)(34,174,123,100)(35,173,124,99)(36,172,125,98)(37,171,126,97)(38,170,127,96)(39,169,128,95)(40,196,129,94)(41,195,130,93)(42,194,131,92)(43,193,132,91)(44,192,133,90)(45,191,134,89)(46,190,135,88)(47,189,136,87)(48,188,137,86)(49,187,138,85)(50,186,139,112)(51,185,140,111)(52,184,113,110)(53,183,114,109)(54,182,115,108)(55,181,116,107)(56,180,117,106), (1,42,15,56)(2,41,16,55)(3,40,17,54)(4,39,18,53)(5,38,19,52)(6,37,20,51)(7,36,21,50)(8,35,22,49)(9,34,23,48)(10,33,24,47)(11,32,25,46)(12,31,26,45)(13,30,27,44)(14,29,28,43)(57,120,71,134)(58,119,72,133)(59,118,73,132)(60,117,74,131)(61,116,75,130)(62,115,76,129)(63,114,77,128)(64,113,78,127)(65,140,79,126)(66,139,80,125)(67,138,81,124)(68,137,82,123)(69,136,83,122)(70,135,84,121)(85,161,99,147)(86,160,100,146)(87,159,101,145)(88,158,102,144)(89,157,103,143)(90,156,104,142)(91,155,105,141)(92,154,106,168)(93,153,107,167)(94,152,108,166)(95,151,109,165)(96,150,110,164)(97,149,111,163)(98,148,112,162)(169,210,183,224)(170,209,184,223)(171,208,185,222)(172,207,186,221)(173,206,187,220)(174,205,188,219)(175,204,189,218)(176,203,190,217)(177,202,191,216)(178,201,192,215)(179,200,193,214)(180,199,194,213)(181,198,195,212)(182,197,196,211) );
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)], [(1,161,74,220),(2,160,75,219),(3,159,76,218),(4,158,77,217),(5,157,78,216),(6,156,79,215),(7,155,80,214),(8,154,81,213),(9,153,82,212),(10,152,83,211),(11,151,84,210),(12,150,57,209),(13,149,58,208),(14,148,59,207),(15,147,60,206),(16,146,61,205),(17,145,62,204),(18,144,63,203),(19,143,64,202),(20,142,65,201),(21,141,66,200),(22,168,67,199),(23,167,68,198),(24,166,69,197),(25,165,70,224),(26,164,71,223),(27,163,72,222),(28,162,73,221),(29,179,118,105),(30,178,119,104),(31,177,120,103),(32,176,121,102),(33,175,122,101),(34,174,123,100),(35,173,124,99),(36,172,125,98),(37,171,126,97),(38,170,127,96),(39,169,128,95),(40,196,129,94),(41,195,130,93),(42,194,131,92),(43,193,132,91),(44,192,133,90),(45,191,134,89),(46,190,135,88),(47,189,136,87),(48,188,137,86),(49,187,138,85),(50,186,139,112),(51,185,140,111),(52,184,113,110),(53,183,114,109),(54,182,115,108),(55,181,116,107),(56,180,117,106)], [(1,42,15,56),(2,41,16,55),(3,40,17,54),(4,39,18,53),(5,38,19,52),(6,37,20,51),(7,36,21,50),(8,35,22,49),(9,34,23,48),(10,33,24,47),(11,32,25,46),(12,31,26,45),(13,30,27,44),(14,29,28,43),(57,120,71,134),(58,119,72,133),(59,118,73,132),(60,117,74,131),(61,116,75,130),(62,115,76,129),(63,114,77,128),(64,113,78,127),(65,140,79,126),(66,139,80,125),(67,138,81,124),(68,137,82,123),(69,136,83,122),(70,135,84,121),(85,161,99,147),(86,160,100,146),(87,159,101,145),(88,158,102,144),(89,157,103,143),(90,156,104,142),(91,155,105,141),(92,154,106,168),(93,153,107,167),(94,152,108,166),(95,151,109,165),(96,150,110,164),(97,149,111,163),(98,148,112,162),(169,210,183,224),(170,209,184,223),(171,208,185,222),(172,207,186,221),(173,206,187,220),(174,205,188,219),(175,204,189,218),(176,203,190,217),(177,202,191,216),(178,201,192,215),(179,200,193,214),(180,199,194,213),(181,198,195,212),(182,197,196,211)]])
C28.44D4 is a maximal subgroup of
C28.14Q16 C4×C56⋊C2 C42.264D14 C4×Dic28 C42.14D14 C42.16D14 C42.20D14 Dic28⋊C4 C23.34D28 C23.35D28 C23.10D28 D28.32D4 D28⋊14D4 Dic14⋊14D4 C22⋊Dic28 D4.D7⋊C4 Dic7⋊6SD16 C28⋊Q8⋊C2 (C8×Dic7)⋊C2 D4⋊(C4×D7) D4⋊2D7⋊C4 D14⋊SD16 C7⋊C8⋊1D4 C7⋊Q16⋊C4 Dic7⋊4Q16 Dic7.1Q16 C56⋊C4.C2 D7×Q8⋊C4 (Q8×D7)⋊C4 D14⋊Q16 C7⋊C8.D4 Dic14.3Q8 C28⋊SD16 D28.19D4 C42.36D14 C4⋊Dic28 C28.7Q16 Dic14⋊4Q8 Dic14⋊Q8 Dic14.Q8 D14.2SD16 C28.(C4○D4) Dic14⋊2Q8 Dic14.2Q8 D14.2Q16 C2.D8⋊7D7 C23.23D28 C56⋊30D4 C56.82D4 C23.46D28 C23.49D28 C56⋊2D4 C56.4D4 (C2×D8).D7 Dic14⋊D4 Dic7⋊3SD16 (C7×Q8).D4 Dic14⋊7D4 Dic14.16D4 Dic7⋊3Q16 D14⋊5Q16
C28.44D4 is a maximal quotient of
C4.8Dic28 C23.30D28 C4.Dic28 C28.47D8 C28.9C42
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | SD16 | Q16 | D14 | C4×D7 | C7⋊D4 | D28 | C56⋊C2 | Dic28 |
| kernel | C28.44D4 | C4⋊Dic7 | C2×C56 | C2×Dic14 | Dic14 | C28 | C2×C14 | C2×C8 | C14 | C14 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C28.44D4 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 9 | 13 |
| 0 | 100 | 19 |
| 15 | 0 | 0 |
| 0 | 35 | 74 |
| 0 | 14 | 78 |
| 1 | 0 | 0 |
| 0 | 99 | 76 |
| 0 | 100 | 14 |
G:=sub<GL(3,GF(113))| [112,0,0,0,9,100,0,13,19],[15,0,0,0,35,14,0,74,78],[1,0,0,0,99,100,0,76,14] >;
C28.44D4 in GAP, Magma, Sage, TeX
C_{28}._{44}D_4 % in TeX
G:=Group("C28.44D4"); // GroupNames label
G:=SmallGroup(224,22);
// by ID
G=gap.SmallGroup(224,22);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,73,79,362,86,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=1,c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^21*b^-1>;
// generators/relations
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