metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊2Q8, C4.4D28, C4⋊2Dic14, C28.27D4, C42.4D7, C7⋊1(C4⋊Q8), (C4×C28).2C2, C14.1(C2×D4), C2.4(C2×D28), C14.2(C2×Q8), (C2×C4).73D14, C4⋊Dic7.4C2, C2.4(C2×Dic14), (C2×C14).10C23, (C2×C28).85C22, (C2×Dic14).2C2, (C2×Dic7).1C22, C22.34(C22×D7), SmallGroup(224,64)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊2Q8
G = < a,b,c | a28=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 246 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×Q8, Dic7, C28, C2×C14, C4⋊Q8, Dic14, C2×Dic7, C2×C28, C2×C28, C4⋊Dic7, C4×C28, C2×Dic14, C28⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, Dic14, D28, C22×D7, C2×Dic14, C2×D28, C28⋊2Q8
►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)
(1 155 98 52)(2 156 99 53)(3 157 100 54)(4 158 101 55)(5 159 102 56)(6 160 103 29)(7 161 104 30)(8 162 105 31)(9 163 106 32)(10 164 107 33)(11 165 108 34)(12 166 109 35)(13 167 110 36)(14 168 111 37)(15 141 112 38)(16 142 85 39)(17 143 86 40)(18 144 87 41)(19 145 88 42)(20 146 89 43)(21 147 90 44)(22 148 91 45)(23 149 92 46)(24 150 93 47)(25 151 94 48)(26 152 95 49)(27 153 96 50)(28 154 97 51)(57 173 140 214)(58 174 113 215)(59 175 114 216)(60 176 115 217)(61 177 116 218)(62 178 117 219)(63 179 118 220)(64 180 119 221)(65 181 120 222)(66 182 121 223)(67 183 122 224)(68 184 123 197)(69 185 124 198)(70 186 125 199)(71 187 126 200)(72 188 127 201)(73 189 128 202)(74 190 129 203)(75 191 130 204)(76 192 131 205)(77 193 132 206)(78 194 133 207)(79 195 134 208)(80 196 135 209)(81 169 136 210)(82 170 137 211)(83 171 138 212)(84 172 139 213)
(1 180 98 221)(2 179 99 220)(3 178 100 219)(4 177 101 218)(5 176 102 217)(6 175 103 216)(7 174 104 215)(8 173 105 214)(9 172 106 213)(10 171 107 212)(11 170 108 211)(12 169 109 210)(13 196 110 209)(14 195 111 208)(15 194 112 207)(16 193 85 206)(17 192 86 205)(18 191 87 204)(19 190 88 203)(20 189 89 202)(21 188 90 201)(22 187 91 200)(23 186 92 199)(24 185 93 198)(25 184 94 197)(26 183 95 224)(27 182 96 223)(28 181 97 222)(29 114 160 59)(30 113 161 58)(31 140 162 57)(32 139 163 84)(33 138 164 83)(34 137 165 82)(35 136 166 81)(36 135 167 80)(37 134 168 79)(38 133 141 78)(39 132 142 77)(40 131 143 76)(41 130 144 75)(42 129 145 74)(43 128 146 73)(44 127 147 72)(45 126 148 71)(46 125 149 70)(47 124 150 69)(48 123 151 68)(49 122 152 67)(50 121 153 66)(51 120 154 65)(52 119 155 64)(53 118 156 63)(54 117 157 62)(55 116 158 61)(56 115 159 60)
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), (1,155,98,52)(2,156,99,53)(3,157,100,54)(4,158,101,55)(5,159,102,56)(6,160,103,29)(7,161,104,30)(8,162,105,31)(9,163,106,32)(10,164,107,33)(11,165,108,34)(12,166,109,35)(13,167,110,36)(14,168,111,37)(15,141,112,38)(16,142,85,39)(17,143,86,40)(18,144,87,41)(19,145,88,42)(20,146,89,43)(21,147,90,44)(22,148,91,45)(23,149,92,46)(24,150,93,47)(25,151,94,48)(26,152,95,49)(27,153,96,50)(28,154,97,51)(57,173,140,214)(58,174,113,215)(59,175,114,216)(60,176,115,217)(61,177,116,218)(62,178,117,219)(63,179,118,220)(64,180,119,221)(65,181,120,222)(66,182,121,223)(67,183,122,224)(68,184,123,197)(69,185,124,198)(70,186,125,199)(71,187,126,200)(72,188,127,201)(73,189,128,202)(74,190,129,203)(75,191,130,204)(76,192,131,205)(77,193,132,206)(78,194,133,207)(79,195,134,208)(80,196,135,209)(81,169,136,210)(82,170,137,211)(83,171,138,212)(84,172,139,213), (1,180,98,221)(2,179,99,220)(3,178,100,219)(4,177,101,218)(5,176,102,217)(6,175,103,216)(7,174,104,215)(8,173,105,214)(9,172,106,213)(10,171,107,212)(11,170,108,211)(12,169,109,210)(13,196,110,209)(14,195,111,208)(15,194,112,207)(16,193,85,206)(17,192,86,205)(18,191,87,204)(19,190,88,203)(20,189,89,202)(21,188,90,201)(22,187,91,200)(23,186,92,199)(24,185,93,198)(25,184,94,197)(26,183,95,224)(27,182,96,223)(28,181,97,222)(29,114,160,59)(30,113,161,58)(31,140,162,57)(32,139,163,84)(33,138,164,83)(34,137,165,82)(35,136,166,81)(36,135,167,80)(37,134,168,79)(38,133,141,78)(39,132,142,77)(40,131,143,76)(41,130,144,75)(42,129,145,74)(43,128,146,73)(44,127,147,72)(45,126,148,71)(46,125,149,70)(47,124,150,69)(48,123,151,68)(49,122,152,67)(50,121,153,66)(51,120,154,65)(52,119,155,64)(53,118,156,63)(54,117,157,62)(55,116,158,61)(56,115,159,60)>;
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), (1,155,98,52)(2,156,99,53)(3,157,100,54)(4,158,101,55)(5,159,102,56)(6,160,103,29)(7,161,104,30)(8,162,105,31)(9,163,106,32)(10,164,107,33)(11,165,108,34)(12,166,109,35)(13,167,110,36)(14,168,111,37)(15,141,112,38)(16,142,85,39)(17,143,86,40)(18,144,87,41)(19,145,88,42)(20,146,89,43)(21,147,90,44)(22,148,91,45)(23,149,92,46)(24,150,93,47)(25,151,94,48)(26,152,95,49)(27,153,96,50)(28,154,97,51)(57,173,140,214)(58,174,113,215)(59,175,114,216)(60,176,115,217)(61,177,116,218)(62,178,117,219)(63,179,118,220)(64,180,119,221)(65,181,120,222)(66,182,121,223)(67,183,122,224)(68,184,123,197)(69,185,124,198)(70,186,125,199)(71,187,126,200)(72,188,127,201)(73,189,128,202)(74,190,129,203)(75,191,130,204)(76,192,131,205)(77,193,132,206)(78,194,133,207)(79,195,134,208)(80,196,135,209)(81,169,136,210)(82,170,137,211)(83,171,138,212)(84,172,139,213), (1,180,98,221)(2,179,99,220)(3,178,100,219)(4,177,101,218)(5,176,102,217)(6,175,103,216)(7,174,104,215)(8,173,105,214)(9,172,106,213)(10,171,107,212)(11,170,108,211)(12,169,109,210)(13,196,110,209)(14,195,111,208)(15,194,112,207)(16,193,85,206)(17,192,86,205)(18,191,87,204)(19,190,88,203)(20,189,89,202)(21,188,90,201)(22,187,91,200)(23,186,92,199)(24,185,93,198)(25,184,94,197)(26,183,95,224)(27,182,96,223)(28,181,97,222)(29,114,160,59)(30,113,161,58)(31,140,162,57)(32,139,163,84)(33,138,164,83)(34,137,165,82)(35,136,166,81)(36,135,167,80)(37,134,168,79)(38,133,141,78)(39,132,142,77)(40,131,143,76)(41,130,144,75)(42,129,145,74)(43,128,146,73)(44,127,147,72)(45,126,148,71)(46,125,149,70)(47,124,150,69)(48,123,151,68)(49,122,152,67)(50,121,153,66)(51,120,154,65)(52,119,155,64)(53,118,156,63)(54,117,157,62)(55,116,158,61)(56,115,159,60) );
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)], [(1,155,98,52),(2,156,99,53),(3,157,100,54),(4,158,101,55),(5,159,102,56),(6,160,103,29),(7,161,104,30),(8,162,105,31),(9,163,106,32),(10,164,107,33),(11,165,108,34),(12,166,109,35),(13,167,110,36),(14,168,111,37),(15,141,112,38),(16,142,85,39),(17,143,86,40),(18,144,87,41),(19,145,88,42),(20,146,89,43),(21,147,90,44),(22,148,91,45),(23,149,92,46),(24,150,93,47),(25,151,94,48),(26,152,95,49),(27,153,96,50),(28,154,97,51),(57,173,140,214),(58,174,113,215),(59,175,114,216),(60,176,115,217),(61,177,116,218),(62,178,117,219),(63,179,118,220),(64,180,119,221),(65,181,120,222),(66,182,121,223),(67,183,122,224),(68,184,123,197),(69,185,124,198),(70,186,125,199),(71,187,126,200),(72,188,127,201),(73,189,128,202),(74,190,129,203),(75,191,130,204),(76,192,131,205),(77,193,132,206),(78,194,133,207),(79,195,134,208),(80,196,135,209),(81,169,136,210),(82,170,137,211),(83,171,138,212),(84,172,139,213)], [(1,180,98,221),(2,179,99,220),(3,178,100,219),(4,177,101,218),(5,176,102,217),(6,175,103,216),(7,174,104,215),(8,173,105,214),(9,172,106,213),(10,171,107,212),(11,170,108,211),(12,169,109,210),(13,196,110,209),(14,195,111,208),(15,194,112,207),(16,193,85,206),(17,192,86,205),(18,191,87,204),(19,190,88,203),(20,189,89,202),(21,188,90,201),(22,187,91,200),(23,186,92,199),(24,185,93,198),(25,184,94,197),(26,183,95,224),(27,182,96,223),(28,181,97,222),(29,114,160,59),(30,113,161,58),(31,140,162,57),(32,139,163,84),(33,138,164,83),(34,137,165,82),(35,136,166,81),(36,135,167,80),(37,134,168,79),(38,133,141,78),(39,132,142,77),(40,131,143,76),(41,130,144,75),(42,129,145,74),(43,128,146,73),(44,127,147,72),(45,126,148,71),(46,125,149,70),(47,124,150,69),(48,123,151,68),(49,122,152,67),(50,121,153,66),(51,120,154,65),(52,119,155,64),(53,118,156,63),(54,117,157,62),(55,116,158,61),(56,115,159,60)]])
C28⋊2Q8 is a maximal subgroup of
C4.Dic28 C28.47D8 C28.2D8 C56⋊9Q8 C28.14Q16 C56⋊8Q8 C8⋊5D28 C4.5D56 C28⋊4Q16 C8⋊Dic14 C42.14D14 C42.20D14 C8.D28 D4.9D28 C28⋊SD16 D28⋊3Q8 D28⋊4Q8 C4⋊Dic28 C28.7Q16 Dic14⋊4Q8 C28.50D8 C28.38SD16 D4.2D28 C28.48SD16 C28.23Q16 C28⋊7Q16 C42.62D14 C42.65D14 C42.68D14 C42.71D14 C28.16D8 C28⋊4SD16 C28.17D8 C28.SD16 C28.Q16 C28⋊3Q16 C28.11Q16 C42.274D14 C42.276D14 C42.89D14 C42.90D14 C42.92D14 C42.99D14 D4×Dic14 C42.106D14 D4⋊6Dic14 D28⋊24D4 D4⋊6D28 C42.117D14 Q8×Dic14 Dic14⋊10Q8 Q8⋊6Dic14 Q8×D28 D28⋊10Q8 C42.135D14 C42.141D14 C42.144D14 C42.148D14 C42.155D14 C42.165D14 C42.238D14 D7×C4⋊Q8 C42.241D14
C28⋊2Q8 is a maximal quotient of
(C2×Dic7)⋊Q8 C14.(C4⋊Q8) C56⋊9Q8 C56⋊8Q8 C56.13Q8 C8⋊Dic14 C28⋊4(C4⋊C4) (C2×C28)⋊10Q8 C42⋊8Dic7
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + |
| image | C1 | C2 | C2 | C2 | D4 | Q8 | D7 | D14 | Dic14 | D28 |
| kernel | C28⋊2Q8 | C4⋊Dic7 | C4×C28 | C2×Dic14 | C28 | C28 | C42 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 4 | 3 | 9 | 24 | 12 |
Matrix representation of C28⋊2Q8 ►in GL4(𝔽29) generated by
| 28 | 3 | 0 | 0 |
| 26 | 8 | 0 | 0 |
| 0 | 0 | 27 | 24 |
| 0 | 0 | 5 | 12 |
| 2 | 18 | 0 | 0 |
| 11 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 9 | 0 | 0 |
| 17 | 7 | 0 | 0 |
| 0 | 0 | 18 | 2 |
| 0 | 0 | 27 | 11 |
G:=sub<GL(4,GF(29))| [28,26,0,0,3,8,0,0,0,0,27,5,0,0,24,12],[2,11,0,0,18,27,0,0,0,0,1,0,0,0,0,1],[22,17,0,0,9,7,0,0,0,0,18,27,0,0,2,11] >;
C28⋊2Q8 in GAP, Magma, Sage, TeX
C_{28}\rtimes_2Q_8 % in TeX
G:=Group("C28:2Q8"); // GroupNames label
G:=SmallGroup(224,64);
// by ID
G=gap.SmallGroup(224,64);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,103,218,50,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations