metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.6D8, C28.1Q8, C14.3Q16, C4.1Dic14, C7⋊C8⋊1C4, C4⋊C4.1D7, C7⋊1(C2.D8), C28.1(C2×C4), C4.11(C4×D7), C14.2(C4⋊C4), C2.1(D4⋊D7), (C2×C4).33D14, (C2×C14).28D4, C4⋊Dic7.8C2, (C2×C28).8C22, C2.1(C7⋊Q16), C2.3(Dic7⋊C4), C22.12(C7⋊D4), (C2×C7⋊C8).1C2, (C7×C4⋊C4).1C2, SmallGroup(224,13)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C22 — C2×C4 — C4⋊C4 |
Generators and relations for C28.Q8
G = < a,b,c | a28=b4=1, c2=a21b2, bab-1=a15, cac-1=a13, cbc-1=a21b-1 >
(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)
(1 142 182 62)(2 157 183 77)(3 144 184 64)(4 159 185 79)(5 146 186 66)(6 161 187 81)(7 148 188 68)(8 163 189 83)(9 150 190 70)(10 165 191 57)(11 152 192 72)(12 167 193 59)(13 154 194 74)(14 141 195 61)(15 156 196 76)(16 143 169 63)(17 158 170 78)(18 145 171 65)(19 160 172 80)(20 147 173 67)(21 162 174 82)(22 149 175 69)(23 164 176 84)(24 151 177 71)(25 166 178 58)(26 153 179 73)(27 168 180 60)(28 155 181 75)(29 136 213 94)(30 123 214 109)(31 138 215 96)(32 125 216 111)(33 140 217 98)(34 127 218 85)(35 114 219 100)(36 129 220 87)(37 116 221 102)(38 131 222 89)(39 118 223 104)(40 133 224 91)(41 120 197 106)(42 135 198 93)(43 122 199 108)(44 137 200 95)(45 124 201 110)(46 139 202 97)(47 126 203 112)(48 113 204 99)(49 128 205 86)(50 115 206 101)(51 130 207 88)(52 117 208 103)(53 132 209 90)(54 119 210 105)(55 134 211 92)(56 121 212 107)
(1 95 175 130 15 109 189 116)(2 108 176 115 16 94 190 129)(3 93 177 128 17 107 191 114)(4 106 178 113 18 92 192 127)(5 91 179 126 19 105 193 140)(6 104 180 139 20 90 194 125)(7 89 181 124 21 103 195 138)(8 102 182 137 22 88 196 123)(9 87 183 122 23 101 169 136)(10 100 184 135 24 86 170 121)(11 85 185 120 25 99 171 134)(12 98 186 133 26 112 172 119)(13 111 187 118 27 97 173 132)(14 96 188 131 28 110 174 117)(29 143 206 84 43 157 220 70)(30 156 207 69 44 142 221 83)(31 141 208 82 45 155 222 68)(32 154 209 67 46 168 223 81)(33 167 210 80 47 153 224 66)(34 152 211 65 48 166 197 79)(35 165 212 78 49 151 198 64)(36 150 213 63 50 164 199 77)(37 163 214 76 51 149 200 62)(38 148 215 61 52 162 201 75)(39 161 216 74 53 147 202 60)(40 146 217 59 54 160 203 73)(41 159 218 72 55 145 204 58)(42 144 219 57 56 158 205 71)
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), (1,142,182,62)(2,157,183,77)(3,144,184,64)(4,159,185,79)(5,146,186,66)(6,161,187,81)(7,148,188,68)(8,163,189,83)(9,150,190,70)(10,165,191,57)(11,152,192,72)(12,167,193,59)(13,154,194,74)(14,141,195,61)(15,156,196,76)(16,143,169,63)(17,158,170,78)(18,145,171,65)(19,160,172,80)(20,147,173,67)(21,162,174,82)(22,149,175,69)(23,164,176,84)(24,151,177,71)(25,166,178,58)(26,153,179,73)(27,168,180,60)(28,155,181,75)(29,136,213,94)(30,123,214,109)(31,138,215,96)(32,125,216,111)(33,140,217,98)(34,127,218,85)(35,114,219,100)(36,129,220,87)(37,116,221,102)(38,131,222,89)(39,118,223,104)(40,133,224,91)(41,120,197,106)(42,135,198,93)(43,122,199,108)(44,137,200,95)(45,124,201,110)(46,139,202,97)(47,126,203,112)(48,113,204,99)(49,128,205,86)(50,115,206,101)(51,130,207,88)(52,117,208,103)(53,132,209,90)(54,119,210,105)(55,134,211,92)(56,121,212,107), (1,95,175,130,15,109,189,116)(2,108,176,115,16,94,190,129)(3,93,177,128,17,107,191,114)(4,106,178,113,18,92,192,127)(5,91,179,126,19,105,193,140)(6,104,180,139,20,90,194,125)(7,89,181,124,21,103,195,138)(8,102,182,137,22,88,196,123)(9,87,183,122,23,101,169,136)(10,100,184,135,24,86,170,121)(11,85,185,120,25,99,171,134)(12,98,186,133,26,112,172,119)(13,111,187,118,27,97,173,132)(14,96,188,131,28,110,174,117)(29,143,206,84,43,157,220,70)(30,156,207,69,44,142,221,83)(31,141,208,82,45,155,222,68)(32,154,209,67,46,168,223,81)(33,167,210,80,47,153,224,66)(34,152,211,65,48,166,197,79)(35,165,212,78,49,151,198,64)(36,150,213,63,50,164,199,77)(37,163,214,76,51,149,200,62)(38,148,215,61,52,162,201,75)(39,161,216,74,53,147,202,60)(40,146,217,59,54,160,203,73)(41,159,218,72,55,145,204,58)(42,144,219,57,56,158,205,71)>;
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), (1,142,182,62)(2,157,183,77)(3,144,184,64)(4,159,185,79)(5,146,186,66)(6,161,187,81)(7,148,188,68)(8,163,189,83)(9,150,190,70)(10,165,191,57)(11,152,192,72)(12,167,193,59)(13,154,194,74)(14,141,195,61)(15,156,196,76)(16,143,169,63)(17,158,170,78)(18,145,171,65)(19,160,172,80)(20,147,173,67)(21,162,174,82)(22,149,175,69)(23,164,176,84)(24,151,177,71)(25,166,178,58)(26,153,179,73)(27,168,180,60)(28,155,181,75)(29,136,213,94)(30,123,214,109)(31,138,215,96)(32,125,216,111)(33,140,217,98)(34,127,218,85)(35,114,219,100)(36,129,220,87)(37,116,221,102)(38,131,222,89)(39,118,223,104)(40,133,224,91)(41,120,197,106)(42,135,198,93)(43,122,199,108)(44,137,200,95)(45,124,201,110)(46,139,202,97)(47,126,203,112)(48,113,204,99)(49,128,205,86)(50,115,206,101)(51,130,207,88)(52,117,208,103)(53,132,209,90)(54,119,210,105)(55,134,211,92)(56,121,212,107), (1,95,175,130,15,109,189,116)(2,108,176,115,16,94,190,129)(3,93,177,128,17,107,191,114)(4,106,178,113,18,92,192,127)(5,91,179,126,19,105,193,140)(6,104,180,139,20,90,194,125)(7,89,181,124,21,103,195,138)(8,102,182,137,22,88,196,123)(9,87,183,122,23,101,169,136)(10,100,184,135,24,86,170,121)(11,85,185,120,25,99,171,134)(12,98,186,133,26,112,172,119)(13,111,187,118,27,97,173,132)(14,96,188,131,28,110,174,117)(29,143,206,84,43,157,220,70)(30,156,207,69,44,142,221,83)(31,141,208,82,45,155,222,68)(32,154,209,67,46,168,223,81)(33,167,210,80,47,153,224,66)(34,152,211,65,48,166,197,79)(35,165,212,78,49,151,198,64)(36,150,213,63,50,164,199,77)(37,163,214,76,51,149,200,62)(38,148,215,61,52,162,201,75)(39,161,216,74,53,147,202,60)(40,146,217,59,54,160,203,73)(41,159,218,72,55,145,204,58)(42,144,219,57,56,158,205,71) );
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)], [(1,142,182,62),(2,157,183,77),(3,144,184,64),(4,159,185,79),(5,146,186,66),(6,161,187,81),(7,148,188,68),(8,163,189,83),(9,150,190,70),(10,165,191,57),(11,152,192,72),(12,167,193,59),(13,154,194,74),(14,141,195,61),(15,156,196,76),(16,143,169,63),(17,158,170,78),(18,145,171,65),(19,160,172,80),(20,147,173,67),(21,162,174,82),(22,149,175,69),(23,164,176,84),(24,151,177,71),(25,166,178,58),(26,153,179,73),(27,168,180,60),(28,155,181,75),(29,136,213,94),(30,123,214,109),(31,138,215,96),(32,125,216,111),(33,140,217,98),(34,127,218,85),(35,114,219,100),(36,129,220,87),(37,116,221,102),(38,131,222,89),(39,118,223,104),(40,133,224,91),(41,120,197,106),(42,135,198,93),(43,122,199,108),(44,137,200,95),(45,124,201,110),(46,139,202,97),(47,126,203,112),(48,113,204,99),(49,128,205,86),(50,115,206,101),(51,130,207,88),(52,117,208,103),(53,132,209,90),(54,119,210,105),(55,134,211,92),(56,121,212,107)], [(1,95,175,130,15,109,189,116),(2,108,176,115,16,94,190,129),(3,93,177,128,17,107,191,114),(4,106,178,113,18,92,192,127),(5,91,179,126,19,105,193,140),(6,104,180,139,20,90,194,125),(7,89,181,124,21,103,195,138),(8,102,182,137,22,88,196,123),(9,87,183,122,23,101,169,136),(10,100,184,135,24,86,170,121),(11,85,185,120,25,99,171,134),(12,98,186,133,26,112,172,119),(13,111,187,118,27,97,173,132),(14,96,188,131,28,110,174,117),(29,143,206,84,43,157,220,70),(30,156,207,69,44,142,221,83),(31,141,208,82,45,155,222,68),(32,154,209,67,46,168,223,81),(33,167,210,80,47,153,224,66),(34,152,211,65,48,166,197,79),(35,165,212,78,49,151,198,64),(36,150,213,63,50,164,199,77),(37,163,214,76,51,149,200,62),(38,148,215,61,52,162,201,75),(39,161,216,74,53,147,202,60),(40,146,217,59,54,160,203,73),(41,159,218,72,55,145,204,58),(42,144,219,57,56,158,205,71)]])
C28.Q8 is a maximal subgroup of
Dic7⋊4D8 D4.D7⋊C4 Dic7.D8 D4.Dic14 D14.D8 D14⋊D8 C8⋊Dic7⋊C2 C7⋊C8⋊1D4 Dic7⋊4Q16 Dic7.Q16 Q8.2Dic14 D14.Q16 C7⋊(C8⋊D4) D14⋊Q16 (C2×C8).D14 Q8⋊D7⋊C4 C56⋊3Q8 Dic14.Q8 C56.8Q8 (C8×D7)⋊C4 C8⋊(C4×D7) C4.Q8⋊D7 C28.(C4○D4) D28.Q8 C56⋊2Q8 Dic14⋊2Q8 C56⋊4Q8 D7×C2.D8 C56⋊(C2×C4) D14.5D8 D14.2Q16 D28⋊2Q8 C4.Dic7⋊C4 (C2×C14).40D8 C4⋊C4.230D14 C28.(C2×Q8) C4⋊C4.233D14 C28.45(C4⋊C4) C4⋊C4.236D14 C28.50D8 D4.3Dic14 C4×D4⋊D7 C42.51D14 C28.23Q16 Q8.3Dic14 C42.56D14 C4×C7⋊Q16 (C2×C14).D8 (C2×D4).D14 C7⋊C8⋊22D4 C7⋊C8⋊5D4 (C2×C14).Q16 C14.(C4○D8) C7⋊C8⋊6D4 C7⋊C8.29D4 Dic14.4Q8 C42.215D14 C42.68D14 D28.4Q8 C28.17D8 C42.76D14 D28⋊6Q8 Dic14⋊5Q8
C28.Q8 is a maximal quotient of
C28.53D8 C8.4Dic14 C8.5Dic14 C8.7Dic14 C28.C42
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R |
order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | - | + | + | + | - | + | - | + | - | |||
image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | D8 | Q16 | D14 | Dic14 | C4×D7 | C7⋊D4 | D4⋊D7 | C7⋊Q16 |
kernel | C28.Q8 | C2×C7⋊C8 | C4⋊Dic7 | C7×C4⋊C4 | C7⋊C8 | C28 | C2×C14 | C4⋊C4 | C14 | C14 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 3 | 3 |
Matrix representation of C28.Q8 ►in GL4(𝔽113) generated by
80 | 9 | 0 | 0 |
104 | 112 | 0 | 0 |
0 | 0 | 0 | 1 |
0 | 0 | 112 | 0 |
55 | 75 | 0 | 0 |
38 | 58 | 0 | 0 |
0 | 0 | 63 | 103 |
0 | 0 | 103 | 50 |
20 | 46 | 0 | 0 |
92 | 93 | 0 | 0 |
0 | 0 | 31 | 31 |
0 | 0 | 82 | 31 |
G:=sub<GL(4,GF(113))| [80,104,0,0,9,112,0,0,0,0,0,112,0,0,1,0],[55,38,0,0,75,58,0,0,0,0,63,103,0,0,103,50],[20,92,0,0,46,93,0,0,0,0,31,82,0,0,31,31] >;
C28.Q8 in GAP, Magma, Sage, TeX
C_{28}.Q_8
% in TeX
G:=Group("C28.Q8");
// GroupNames label
G:=SmallGroup(224,13);
// by ID
G=gap.SmallGroup(224,13);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,121,31,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=1,c^2=a^21*b^2,b*a*b^-1=a^15,c*a*c^-1=a^13,c*b*c^-1=a^21*b^-1>;
// generators/relations
Export