metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D56, C28.30D8, C28.26SD16, C42.261D14, (C4×C8)⋊5D7, (C4×C56)⋊5C2, C2.5(C2×D56), C14.3(C2×D8), C28⋊2Q8⋊2C2, (C2×C4).80D28, C7⋊1(C4.4D8), C2.D56⋊1C2, (C2×C8).287D14, (C2×C28).377D4, C28⋊4D4.2C2, C4.5(C56⋊C2), C14.5(C2×SD16), (C2×D28).2C22, C22.90(C2×D28), C4⋊Dic7.5C22, C4.102(C4○D28), C28.218(C4○D4), (C2×C28).723C23, (C4×C28).307C22, (C2×C56).347C22, C14.5(C4.4D4), C2.10(C4.D28), C2.8(C2×C56⋊C2), (C2×C14).106(C2×D4), (C2×C4).666(C22×D7), SmallGroup(448,228)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.5D56
G = < a,b,c | a4=b56=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 900 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C56, Dic14, D28, C2×Dic7, C2×C28, C22×D7, C4.4D8, C4⋊Dic7, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C2×D28, C2×D28, C2.D56, C4×C56, C28⋊2Q8, C28⋊4D4, C4.5D56
Quotients: C1, C2, C22, D4, C23, D7, D8, SD16, C2×D4, C4○D4, D14, C4.4D4, C2×D8, C2×SD16, D28, C22×D7, C4.4D8, C56⋊C2, D56, C2×D28, C4○D28, C4.D28, C2×C56⋊C2, C2×D56, C4.5D56
►On 224 points
Generators in S224
(1 202 144 69)(2 203 145 70)(3 204 146 71)(4 205 147 72)(5 206 148 73)(6 207 149 74)(7 208 150 75)(8 209 151 76)(9 210 152 77)(10 211 153 78)(11 212 154 79)(12 213 155 80)(13 214 156 81)(14 215 157 82)(15 216 158 83)(16 217 159 84)(17 218 160 85)(18 219 161 86)(19 220 162 87)(20 221 163 88)(21 222 164 89)(22 223 165 90)(23 224 166 91)(24 169 167 92)(25 170 168 93)(26 171 113 94)(27 172 114 95)(28 173 115 96)(29 174 116 97)(30 175 117 98)(31 176 118 99)(32 177 119 100)(33 178 120 101)(34 179 121 102)(35 180 122 103)(36 181 123 104)(37 182 124 105)(38 183 125 106)(39 184 126 107)(40 185 127 108)(41 186 128 109)(42 187 129 110)(43 188 130 111)(44 189 131 112)(45 190 132 57)(46 191 133 58)(47 192 134 59)(48 193 135 60)(49 194 136 61)(50 195 137 62)(51 196 138 63)(52 197 139 64)(53 198 140 65)(54 199 141 66)(55 200 142 67)(56 201 143 68)
(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)
(1 110 144 187)(2 186 145 109)(3 108 146 185)(4 184 147 107)(5 106 148 183)(6 182 149 105)(7 104 150 181)(8 180 151 103)(9 102 152 179)(10 178 153 101)(11 100 154 177)(12 176 155 99)(13 98 156 175)(14 174 157 97)(15 96 158 173)(16 172 159 95)(17 94 160 171)(18 170 161 93)(19 92 162 169)(20 224 163 91)(21 90 164 223)(22 222 165 89)(23 88 166 221)(24 220 167 87)(25 86 168 219)(26 218 113 85)(27 84 114 217)(28 216 115 83)(29 82 116 215)(30 214 117 81)(31 80 118 213)(32 212 119 79)(33 78 120 211)(34 210 121 77)(35 76 122 209)(36 208 123 75)(37 74 124 207)(38 206 125 73)(39 72 126 205)(40 204 127 71)(41 70 128 203)(42 202 129 69)(43 68 130 201)(44 200 131 67)(45 66 132 199)(46 198 133 65)(47 64 134 197)(48 196 135 63)(49 62 136 195)(50 194 137 61)(51 60 138 193)(52 192 139 59)(53 58 140 191)(54 190 141 57)(55 112 142 189)(56 188 143 111)
G:=sub<Sym(224)| (1,202,144,69)(2,203,145,70)(3,204,146,71)(4,205,147,72)(5,206,148,73)(6,207,149,74)(7,208,150,75)(8,209,151,76)(9,210,152,77)(10,211,153,78)(11,212,154,79)(12,213,155,80)(13,214,156,81)(14,215,157,82)(15,216,158,83)(16,217,159,84)(17,218,160,85)(18,219,161,86)(19,220,162,87)(20,221,163,88)(21,222,164,89)(22,223,165,90)(23,224,166,91)(24,169,167,92)(25,170,168,93)(26,171,113,94)(27,172,114,95)(28,173,115,96)(29,174,116,97)(30,175,117,98)(31,176,118,99)(32,177,119,100)(33,178,120,101)(34,179,121,102)(35,180,122,103)(36,181,123,104)(37,182,124,105)(38,183,125,106)(39,184,126,107)(40,185,127,108)(41,186,128,109)(42,187,129,110)(43,188,130,111)(44,189,131,112)(45,190,132,57)(46,191,133,58)(47,192,134,59)(48,193,135,60)(49,194,136,61)(50,195,137,62)(51,196,138,63)(52,197,139,64)(53,198,140,65)(54,199,141,66)(55,200,142,67)(56,201,143,68), (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), (1,110,144,187)(2,186,145,109)(3,108,146,185)(4,184,147,107)(5,106,148,183)(6,182,149,105)(7,104,150,181)(8,180,151,103)(9,102,152,179)(10,178,153,101)(11,100,154,177)(12,176,155,99)(13,98,156,175)(14,174,157,97)(15,96,158,173)(16,172,159,95)(17,94,160,171)(18,170,161,93)(19,92,162,169)(20,224,163,91)(21,90,164,223)(22,222,165,89)(23,88,166,221)(24,220,167,87)(25,86,168,219)(26,218,113,85)(27,84,114,217)(28,216,115,83)(29,82,116,215)(30,214,117,81)(31,80,118,213)(32,212,119,79)(33,78,120,211)(34,210,121,77)(35,76,122,209)(36,208,123,75)(37,74,124,207)(38,206,125,73)(39,72,126,205)(40,204,127,71)(41,70,128,203)(42,202,129,69)(43,68,130,201)(44,200,131,67)(45,66,132,199)(46,198,133,65)(47,64,134,197)(48,196,135,63)(49,62,136,195)(50,194,137,61)(51,60,138,193)(52,192,139,59)(53,58,140,191)(54,190,141,57)(55,112,142,189)(56,188,143,111)>;
G:=Group( (1,202,144,69)(2,203,145,70)(3,204,146,71)(4,205,147,72)(5,206,148,73)(6,207,149,74)(7,208,150,75)(8,209,151,76)(9,210,152,77)(10,211,153,78)(11,212,154,79)(12,213,155,80)(13,214,156,81)(14,215,157,82)(15,216,158,83)(16,217,159,84)(17,218,160,85)(18,219,161,86)(19,220,162,87)(20,221,163,88)(21,222,164,89)(22,223,165,90)(23,224,166,91)(24,169,167,92)(25,170,168,93)(26,171,113,94)(27,172,114,95)(28,173,115,96)(29,174,116,97)(30,175,117,98)(31,176,118,99)(32,177,119,100)(33,178,120,101)(34,179,121,102)(35,180,122,103)(36,181,123,104)(37,182,124,105)(38,183,125,106)(39,184,126,107)(40,185,127,108)(41,186,128,109)(42,187,129,110)(43,188,130,111)(44,189,131,112)(45,190,132,57)(46,191,133,58)(47,192,134,59)(48,193,135,60)(49,194,136,61)(50,195,137,62)(51,196,138,63)(52,197,139,64)(53,198,140,65)(54,199,141,66)(55,200,142,67)(56,201,143,68), (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), (1,110,144,187)(2,186,145,109)(3,108,146,185)(4,184,147,107)(5,106,148,183)(6,182,149,105)(7,104,150,181)(8,180,151,103)(9,102,152,179)(10,178,153,101)(11,100,154,177)(12,176,155,99)(13,98,156,175)(14,174,157,97)(15,96,158,173)(16,172,159,95)(17,94,160,171)(18,170,161,93)(19,92,162,169)(20,224,163,91)(21,90,164,223)(22,222,165,89)(23,88,166,221)(24,220,167,87)(25,86,168,219)(26,218,113,85)(27,84,114,217)(28,216,115,83)(29,82,116,215)(30,214,117,81)(31,80,118,213)(32,212,119,79)(33,78,120,211)(34,210,121,77)(35,76,122,209)(36,208,123,75)(37,74,124,207)(38,206,125,73)(39,72,126,205)(40,204,127,71)(41,70,128,203)(42,202,129,69)(43,68,130,201)(44,200,131,67)(45,66,132,199)(46,198,133,65)(47,64,134,197)(48,196,135,63)(49,62,136,195)(50,194,137,61)(51,60,138,193)(52,192,139,59)(53,58,140,191)(54,190,141,57)(55,112,142,189)(56,188,143,111) );
G=PermutationGroup([[(1,202,144,69),(2,203,145,70),(3,204,146,71),(4,205,147,72),(5,206,148,73),(6,207,149,74),(7,208,150,75),(8,209,151,76),(9,210,152,77),(10,211,153,78),(11,212,154,79),(12,213,155,80),(13,214,156,81),(14,215,157,82),(15,216,158,83),(16,217,159,84),(17,218,160,85),(18,219,161,86),(19,220,162,87),(20,221,163,88),(21,222,164,89),(22,223,165,90),(23,224,166,91),(24,169,167,92),(25,170,168,93),(26,171,113,94),(27,172,114,95),(28,173,115,96),(29,174,116,97),(30,175,117,98),(31,176,118,99),(32,177,119,100),(33,178,120,101),(34,179,121,102),(35,180,122,103),(36,181,123,104),(37,182,124,105),(38,183,125,106),(39,184,126,107),(40,185,127,108),(41,186,128,109),(42,187,129,110),(43,188,130,111),(44,189,131,112),(45,190,132,57),(46,191,133,58),(47,192,134,59),(48,193,135,60),(49,194,136,61),(50,195,137,62),(51,196,138,63),(52,197,139,64),(53,198,140,65),(54,199,141,66),(55,200,142,67),(56,201,143,68)], [(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)], [(1,110,144,187),(2,186,145,109),(3,108,146,185),(4,184,147,107),(5,106,148,183),(6,182,149,105),(7,104,150,181),(8,180,151,103),(9,102,152,179),(10,178,153,101),(11,100,154,177),(12,176,155,99),(13,98,156,175),(14,174,157,97),(15,96,158,173),(16,172,159,95),(17,94,160,171),(18,170,161,93),(19,92,162,169),(20,224,163,91),(21,90,164,223),(22,222,165,89),(23,88,166,221),(24,220,167,87),(25,86,168,219),(26,218,113,85),(27,84,114,217),(28,216,115,83),(29,82,116,215),(30,214,117,81),(31,80,118,213),(32,212,119,79),(33,78,120,211),(34,210,121,77),(35,76,122,209),(36,208,123,75),(37,74,124,207),(38,206,125,73),(39,72,126,205),(40,204,127,71),(41,70,128,203),(42,202,129,69),(43,68,130,201),(44,200,131,67),(45,66,132,199),(46,198,133,65),(47,64,134,197),(48,196,135,63),(49,62,136,195),(50,194,137,61),(51,60,138,193),(52,192,139,59),(53,58,140,191),(54,190,141,57),(55,112,142,189),(56,188,143,111)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 56 | 2 | ··· | 2 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 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 | C2 | D4 | D7 | D8 | SD16 | C4○D4 | D14 | D14 | D28 | C56⋊C2 | D56 | C4○D28 |
| kernel | C4.5D56 | C2.D56 | C4×C56 | C28⋊2Q8 | C28⋊4D4 | C2×C28 | C4×C8 | C28 | C28 | C28 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 3 | 4 | 4 | 4 | 3 | 6 | 12 | 24 | 24 | 24 |
Matrix representation of C4.5D56 ►in GL4(𝔽113) generated by
| 58 | 75 | 0 | 0 |
| 38 | 55 | 0 | 0 |
| 0 | 0 | 112 | 7 |
| 0 | 0 | 32 | 1 |
| 22 | 98 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 91 |
| 0 | 0 | 77 | 26 |
| 28 | 34 | 0 | 0 |
| 60 | 85 | 0 | 0 |
| 0 | 0 | 0 | 91 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(113))| [58,38,0,0,75,55,0,0,0,0,112,32,0,0,7,1],[22,15,0,0,98,0,0,0,0,0,0,77,0,0,91,26],[28,60,0,0,34,85,0,0,0,0,0,36,0,0,91,0] >;
C4.5D56 in GAP, Magma, Sage, TeX
C_4._5D_{56} % in TeX
G:=Group("C4.5D56"); // GroupNames label
G:=SmallGroup(448,228);
// by ID
G=gap.SmallGroup(448,228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations