metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.36D4, (C2×Q16)⋊8D7, (C14×Q16)⋊9C2, C8⋊Dic7⋊22C2, (C2×C8).95D14, C7⋊5(C8.D4), C8.5(C7⋊D4), C28.187(C2×D4), (C2×Q8).65D14, Q8⋊Dic7⋊36C2, D14⋊3Q8.9C2, (C2×Dic7).77D4, (C22×D7).41D4, C22.280(D4×D7), C28.108(C4○D4), C4.37(D4⋊2D7), C2.23(C28⋊2D4), (C2×C56).150C22, (C2×C28).463C23, (Q8×C14).92C22, C14.122(C4⋊D4), C2.30(Q16⋊D7), C14.80(C8.C22), C4⋊Dic7.186C22, C4.86(C2×C7⋊D4), (C2×C8⋊D7).5C2, (C2×C4×D7).54C22, (C2×C14).374(C2×D4), (C2×C7⋊C8).167C22, (C2×C4).551(C22×D7), SmallGroup(448,723)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.36D4
G = < a,b,c | a56=b4=c2=1, bab-1=a27, cac=a13, cbc=a28b-1 >
Subgroups: 516 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic7, C28, C28, D14, C2×C14, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C8.D4, C8⋊D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C2×C56, C7×Q16, C2×C4×D7, Q8×C14, C8⋊Dic7, Q8⋊Dic7, C2×C8⋊D7, D14⋊3Q8, C14×Q16, C56.36D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8.C22, C7⋊D4, C22×D7, C8.D4, D4×D7, D4⋊2D7, C2×C7⋊D4, Q16⋊D7, C28⋊2D4, C56.36D4
►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)
(1 198 114 85)(2 169 115 112)(3 196 116 83)(4 223 117 110)(5 194 118 81)(6 221 119 108)(7 192 120 79)(8 219 121 106)(9 190 122 77)(10 217 123 104)(11 188 124 75)(12 215 125 102)(13 186 126 73)(14 213 127 100)(15 184 128 71)(16 211 129 98)(17 182 130 69)(18 209 131 96)(19 180 132 67)(20 207 133 94)(21 178 134 65)(22 205 135 92)(23 176 136 63)(24 203 137 90)(25 174 138 61)(26 201 139 88)(27 172 140 59)(28 199 141 86)(29 170 142 57)(30 197 143 84)(31 224 144 111)(32 195 145 82)(33 222 146 109)(34 193 147 80)(35 220 148 107)(36 191 149 78)(37 218 150 105)(38 189 151 76)(39 216 152 103)(40 187 153 74)(41 214 154 101)(42 185 155 72)(43 212 156 99)(44 183 157 70)(45 210 158 97)(46 181 159 68)(47 208 160 95)(48 179 161 66)(49 206 162 93)(50 177 163 64)(51 204 164 91)(52 175 165 62)(53 202 166 89)(54 173 167 60)(55 200 168 87)(56 171 113 58)
(2 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 198)(58 211)(59 224)(60 181)(61 194)(62 207)(63 220)(64 177)(65 190)(66 203)(67 216)(68 173)(69 186)(70 199)(71 212)(72 169)(73 182)(74 195)(75 208)(76 221)(77 178)(78 191)(79 204)(80 217)(81 174)(82 187)(83 200)(84 213)(85 170)(86 183)(87 196)(88 209)(89 222)(90 179)(91 192)(92 205)(93 218)(94 175)(95 188)(96 201)(97 214)(98 171)(99 184)(100 197)(101 210)(102 223)(103 180)(104 193)(105 206)(106 219)(107 176)(108 189)(109 202)(110 215)(111 172)(112 185)(113 157)(115 127)(116 140)(117 153)(118 166)(119 123)(120 136)(121 149)(122 162)(124 132)(125 145)(126 158)(129 141)(130 154)(131 167)(133 137)(134 150)(135 163)(138 146)(139 159)(143 155)(144 168)(147 151)(148 164)(152 160)(161 165)
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), (1,198,114,85)(2,169,115,112)(3,196,116,83)(4,223,117,110)(5,194,118,81)(6,221,119,108)(7,192,120,79)(8,219,121,106)(9,190,122,77)(10,217,123,104)(11,188,124,75)(12,215,125,102)(13,186,126,73)(14,213,127,100)(15,184,128,71)(16,211,129,98)(17,182,130,69)(18,209,131,96)(19,180,132,67)(20,207,133,94)(21,178,134,65)(22,205,135,92)(23,176,136,63)(24,203,137,90)(25,174,138,61)(26,201,139,88)(27,172,140,59)(28,199,141,86)(29,170,142,57)(30,197,143,84)(31,224,144,111)(32,195,145,82)(33,222,146,109)(34,193,147,80)(35,220,148,107)(36,191,149,78)(37,218,150,105)(38,189,151,76)(39,216,152,103)(40,187,153,74)(41,214,154,101)(42,185,155,72)(43,212,156,99)(44,183,157,70)(45,210,158,97)(46,181,159,68)(47,208,160,95)(48,179,161,66)(49,206,162,93)(50,177,163,64)(51,204,164,91)(52,175,165,62)(53,202,166,89)(54,173,167,60)(55,200,168,87)(56,171,113,58), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,198)(58,211)(59,224)(60,181)(61,194)(62,207)(63,220)(64,177)(65,190)(66,203)(67,216)(68,173)(69,186)(70,199)(71,212)(72,169)(73,182)(74,195)(75,208)(76,221)(77,178)(78,191)(79,204)(80,217)(81,174)(82,187)(83,200)(84,213)(85,170)(86,183)(87,196)(88,209)(89,222)(90,179)(91,192)(92,205)(93,218)(94,175)(95,188)(96,201)(97,214)(98,171)(99,184)(100,197)(101,210)(102,223)(103,180)(104,193)(105,206)(106,219)(107,176)(108,189)(109,202)(110,215)(111,172)(112,185)(113,157)(115,127)(116,140)(117,153)(118,166)(119,123)(120,136)(121,149)(122,162)(124,132)(125,145)(126,158)(129,141)(130,154)(131,167)(133,137)(134,150)(135,163)(138,146)(139,159)(143,155)(144,168)(147,151)(148,164)(152,160)(161,165)>;
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), (1,198,114,85)(2,169,115,112)(3,196,116,83)(4,223,117,110)(5,194,118,81)(6,221,119,108)(7,192,120,79)(8,219,121,106)(9,190,122,77)(10,217,123,104)(11,188,124,75)(12,215,125,102)(13,186,126,73)(14,213,127,100)(15,184,128,71)(16,211,129,98)(17,182,130,69)(18,209,131,96)(19,180,132,67)(20,207,133,94)(21,178,134,65)(22,205,135,92)(23,176,136,63)(24,203,137,90)(25,174,138,61)(26,201,139,88)(27,172,140,59)(28,199,141,86)(29,170,142,57)(30,197,143,84)(31,224,144,111)(32,195,145,82)(33,222,146,109)(34,193,147,80)(35,220,148,107)(36,191,149,78)(37,218,150,105)(38,189,151,76)(39,216,152,103)(40,187,153,74)(41,214,154,101)(42,185,155,72)(43,212,156,99)(44,183,157,70)(45,210,158,97)(46,181,159,68)(47,208,160,95)(48,179,161,66)(49,206,162,93)(50,177,163,64)(51,204,164,91)(52,175,165,62)(53,202,166,89)(54,173,167,60)(55,200,168,87)(56,171,113,58), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,198)(58,211)(59,224)(60,181)(61,194)(62,207)(63,220)(64,177)(65,190)(66,203)(67,216)(68,173)(69,186)(70,199)(71,212)(72,169)(73,182)(74,195)(75,208)(76,221)(77,178)(78,191)(79,204)(80,217)(81,174)(82,187)(83,200)(84,213)(85,170)(86,183)(87,196)(88,209)(89,222)(90,179)(91,192)(92,205)(93,218)(94,175)(95,188)(96,201)(97,214)(98,171)(99,184)(100,197)(101,210)(102,223)(103,180)(104,193)(105,206)(106,219)(107,176)(108,189)(109,202)(110,215)(111,172)(112,185)(113,157)(115,127)(116,140)(117,153)(118,166)(119,123)(120,136)(121,149)(122,162)(124,132)(125,145)(126,158)(129,141)(130,154)(131,167)(133,137)(134,150)(135,163)(138,146)(139,159)(143,155)(144,168)(147,151)(148,164)(152,160)(161,165) );
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)], [(1,198,114,85),(2,169,115,112),(3,196,116,83),(4,223,117,110),(5,194,118,81),(6,221,119,108),(7,192,120,79),(8,219,121,106),(9,190,122,77),(10,217,123,104),(11,188,124,75),(12,215,125,102),(13,186,126,73),(14,213,127,100),(15,184,128,71),(16,211,129,98),(17,182,130,69),(18,209,131,96),(19,180,132,67),(20,207,133,94),(21,178,134,65),(22,205,135,92),(23,176,136,63),(24,203,137,90),(25,174,138,61),(26,201,139,88),(27,172,140,59),(28,199,141,86),(29,170,142,57),(30,197,143,84),(31,224,144,111),(32,195,145,82),(33,222,146,109),(34,193,147,80),(35,220,148,107),(36,191,149,78),(37,218,150,105),(38,189,151,76),(39,216,152,103),(40,187,153,74),(41,214,154,101),(42,185,155,72),(43,212,156,99),(44,183,157,70),(45,210,158,97),(46,181,159,68),(47,208,160,95),(48,179,161,66),(49,206,162,93),(50,177,163,64),(51,204,164,91),(52,175,165,62),(53,202,166,89),(54,173,167,60),(55,200,168,87),(56,171,113,58)], [(2,14),(3,27),(4,40),(5,53),(6,10),(7,23),(8,36),(9,49),(11,19),(12,32),(13,45),(16,28),(17,41),(18,54),(20,24),(21,37),(22,50),(25,33),(26,46),(30,42),(31,55),(34,38),(35,51),(39,47),(44,56),(48,52),(57,198),(58,211),(59,224),(60,181),(61,194),(62,207),(63,220),(64,177),(65,190),(66,203),(67,216),(68,173),(69,186),(70,199),(71,212),(72,169),(73,182),(74,195),(75,208),(76,221),(77,178),(78,191),(79,204),(80,217),(81,174),(82,187),(83,200),(84,213),(85,170),(86,183),(87,196),(88,209),(89,222),(90,179),(91,192),(92,205),(93,218),(94,175),(95,188),(96,201),(97,214),(98,171),(99,184),(100,197),(101,210),(102,223),(103,180),(104,193),(105,206),(106,219),(107,176),(108,189),(109,202),(110,215),(111,172),(112,185),(113,157),(115,127),(116,140),(117,153),(118,166),(119,123),(120,136),(121,149),(122,162),(124,132),(125,145),(126,158),(129,141),(130,154),(131,167),(133,137),(134,150),(135,163),(138,146),(139,159),(143,155),(144,168),(147,151),(148,164),(152,160),(161,165)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 2 | 2 | 8 | 8 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C8.C22 | D4⋊2D7 | D4×D7 | Q16⋊D7 |
| kernel | C56.36D4 | C8⋊Dic7 | Q8⋊Dic7 | C2×C8⋊D7 | D14⋊3Q8 | C14×Q16 | C56 | C2×Dic7 | C22×D7 | C2×Q16 | C28 | C2×C8 | C2×Q8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56.36D4 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 37 | 42 | 37 |
| 0 | 0 | 76 | 36 | 76 | 36 |
| 0 | 0 | 71 | 76 | 42 | 37 |
| 0 | 0 | 37 | 77 | 76 | 36 |
| 39 | 7 | 0 | 0 | 0 | 0 |
| 57 | 74 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 75 | 0 | 0 |
| 0 | 0 | 32 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 38 |
| 0 | 0 | 0 | 0 | 81 | 58 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 5 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,42,76,71,37,0,0,37,36,76,77,0,0,42,76,42,76,0,0,37,36,37,36],[39,57,0,0,0,0,7,74,0,0,0,0,0,0,58,32,0,0,0,0,75,55,0,0,0,0,0,0,55,81,0,0,0,0,38,58],[1,5,0,0,0,0,0,112,0,0,0,0,0,0,1,9,0,0,0,0,0,112,0,0,0,0,0,0,1,9,0,0,0,0,0,112] >;
C56.36D4 in GAP, Magma, Sage, TeX
C_{56}._{36}D_4 % in TeX
G:=Group("C56.36D4"); // GroupNames label
G:=SmallGroup(448,723);
// by ID
G=gap.SmallGroup(448,723);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,254,219,184,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^27,c*a*c=a^13,c*b*c=a^28*b^-1>;
// generators/relations