direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C8○D4, D4.C52, Q8.C52, M4(2)⋊5C26, C52.54C23, C104.30C22, (C2×C8)⋊7C26, C8.7(C2×C26), C4.5(C2×C52), (C2×C104)⋊15C2, C52.53(C2×C4), C4○D4.3C26, (D4×C13).3C4, (Q8×C13).3C4, C2.7(C22×C52), C22.1(C2×C52), C4.12(C22×C26), C26.48(C22×C4), (C13×M4(2))⋊11C2, (C2×C52).128C22, (C2×C26).28(C2×C4), (C2×C4).24(C2×C26), (C13×C4○D4).6C2, SmallGroup(416,192)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C8○D4
G = < a,b,c,d | a13=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, C13, C2×C8, M4(2), C4○D4, C26, C26, C8○D4, C52, C52, C2×C26, C104, C104, C2×C52, D4×C13, Q8×C13, C2×C104, C13×M4(2), C13×C4○D4, C13×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C13, C22×C4, C26, C8○D4, C52, C2×C26, C2×C52, C22×C26, C22×C52, C13×C8○D4
►On 208 points
Generators in S208
(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)
(1 179 127 90 155 93 31 204)(2 180 128 91 156 94 32 205)(3 181 129 79 144 95 33 206)(4 182 130 80 145 96 34 207)(5 170 118 81 146 97 35 208)(6 171 119 82 147 98 36 196)(7 172 120 83 148 99 37 197)(8 173 121 84 149 100 38 198)(9 174 122 85 150 101 39 199)(10 175 123 86 151 102 27 200)(11 176 124 87 152 103 28 201)(12 177 125 88 153 104 29 202)(13 178 126 89 154 92 30 203)(14 73 133 106 184 168 41 65)(15 74 134 107 185 169 42 53)(16 75 135 108 186 157 43 54)(17 76 136 109 187 158 44 55)(18 77 137 110 188 159 45 56)(19 78 138 111 189 160 46 57)(20 66 139 112 190 161 47 58)(21 67 140 113 191 162 48 59)(22 68 141 114 192 163 49 60)(23 69 142 115 193 164 50 61)(24 70 143 116 194 165 51 62)(25 71 131 117 195 166 52 63)(26 72 132 105 183 167 40 64)
(1 192 155 22)(2 193 156 23)(3 194 144 24)(4 195 145 25)(5 183 146 26)(6 184 147 14)(7 185 148 15)(8 186 149 16)(9 187 150 17)(10 188 151 18)(11 189 152 19)(12 190 153 20)(13 191 154 21)(27 137 123 45)(28 138 124 46)(29 139 125 47)(30 140 126 48)(31 141 127 49)(32 142 128 50)(33 143 129 51)(34 131 130 52)(35 132 118 40)(36 133 119 41)(37 134 120 42)(38 135 121 43)(39 136 122 44)(53 197 107 83)(54 198 108 84)(55 199 109 85)(56 200 110 86)(57 201 111 87)(58 202 112 88)(59 203 113 89)(60 204 114 90)(61 205 115 91)(62 206 116 79)(63 207 117 80)(64 208 105 81)(65 196 106 82)(66 177 161 104)(67 178 162 92)(68 179 163 93)(69 180 164 94)(70 181 165 95)(71 182 166 96)(72 170 167 97)(73 171 168 98)(74 172 169 99)(75 173 157 100)(76 174 158 101)(77 175 159 102)(78 176 160 103)
(14 184)(15 185)(16 186)(17 187)(18 188)(19 189)(20 190)(21 191)(22 192)(23 193)(24 194)(25 195)(26 183)(40 132)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 131)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 115)(62 116)(63 117)(64 105)(65 106)(66 161)(67 162)(68 163)(69 164)(70 165)(71 166)(72 167)(73 168)(74 169)(75 157)(76 158)(77 159)(78 160)
G:=sub<Sym(208)| (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), (1,179,127,90,155,93,31,204)(2,180,128,91,156,94,32,205)(3,181,129,79,144,95,33,206)(4,182,130,80,145,96,34,207)(5,170,118,81,146,97,35,208)(6,171,119,82,147,98,36,196)(7,172,120,83,148,99,37,197)(8,173,121,84,149,100,38,198)(9,174,122,85,150,101,39,199)(10,175,123,86,151,102,27,200)(11,176,124,87,152,103,28,201)(12,177,125,88,153,104,29,202)(13,178,126,89,154,92,30,203)(14,73,133,106,184,168,41,65)(15,74,134,107,185,169,42,53)(16,75,135,108,186,157,43,54)(17,76,136,109,187,158,44,55)(18,77,137,110,188,159,45,56)(19,78,138,111,189,160,46,57)(20,66,139,112,190,161,47,58)(21,67,140,113,191,162,48,59)(22,68,141,114,192,163,49,60)(23,69,142,115,193,164,50,61)(24,70,143,116,194,165,51,62)(25,71,131,117,195,166,52,63)(26,72,132,105,183,167,40,64), (1,192,155,22)(2,193,156,23)(3,194,144,24)(4,195,145,25)(5,183,146,26)(6,184,147,14)(7,185,148,15)(8,186,149,16)(9,187,150,17)(10,188,151,18)(11,189,152,19)(12,190,153,20)(13,191,154,21)(27,137,123,45)(28,138,124,46)(29,139,125,47)(30,140,126,48)(31,141,127,49)(32,142,128,50)(33,143,129,51)(34,131,130,52)(35,132,118,40)(36,133,119,41)(37,134,120,42)(38,135,121,43)(39,136,122,44)(53,197,107,83)(54,198,108,84)(55,199,109,85)(56,200,110,86)(57,201,111,87)(58,202,112,88)(59,203,113,89)(60,204,114,90)(61,205,115,91)(62,206,116,79)(63,207,117,80)(64,208,105,81)(65,196,106,82)(66,177,161,104)(67,178,162,92)(68,179,163,93)(69,180,164,94)(70,181,165,95)(71,182,166,96)(72,170,167,97)(73,171,168,98)(74,172,169,99)(75,173,157,100)(76,174,158,101)(77,175,159,102)(78,176,160,103), (14,184)(15,185)(16,186)(17,187)(18,188)(19,189)(20,190)(21,191)(22,192)(23,193)(24,194)(25,195)(26,183)(40,132)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,131)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,105)(65,106)(66,161)(67,162)(68,163)(69,164)(70,165)(71,166)(72,167)(73,168)(74,169)(75,157)(76,158)(77,159)(78,160)>;
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), (1,179,127,90,155,93,31,204)(2,180,128,91,156,94,32,205)(3,181,129,79,144,95,33,206)(4,182,130,80,145,96,34,207)(5,170,118,81,146,97,35,208)(6,171,119,82,147,98,36,196)(7,172,120,83,148,99,37,197)(8,173,121,84,149,100,38,198)(9,174,122,85,150,101,39,199)(10,175,123,86,151,102,27,200)(11,176,124,87,152,103,28,201)(12,177,125,88,153,104,29,202)(13,178,126,89,154,92,30,203)(14,73,133,106,184,168,41,65)(15,74,134,107,185,169,42,53)(16,75,135,108,186,157,43,54)(17,76,136,109,187,158,44,55)(18,77,137,110,188,159,45,56)(19,78,138,111,189,160,46,57)(20,66,139,112,190,161,47,58)(21,67,140,113,191,162,48,59)(22,68,141,114,192,163,49,60)(23,69,142,115,193,164,50,61)(24,70,143,116,194,165,51,62)(25,71,131,117,195,166,52,63)(26,72,132,105,183,167,40,64), (1,192,155,22)(2,193,156,23)(3,194,144,24)(4,195,145,25)(5,183,146,26)(6,184,147,14)(7,185,148,15)(8,186,149,16)(9,187,150,17)(10,188,151,18)(11,189,152,19)(12,190,153,20)(13,191,154,21)(27,137,123,45)(28,138,124,46)(29,139,125,47)(30,140,126,48)(31,141,127,49)(32,142,128,50)(33,143,129,51)(34,131,130,52)(35,132,118,40)(36,133,119,41)(37,134,120,42)(38,135,121,43)(39,136,122,44)(53,197,107,83)(54,198,108,84)(55,199,109,85)(56,200,110,86)(57,201,111,87)(58,202,112,88)(59,203,113,89)(60,204,114,90)(61,205,115,91)(62,206,116,79)(63,207,117,80)(64,208,105,81)(65,196,106,82)(66,177,161,104)(67,178,162,92)(68,179,163,93)(69,180,164,94)(70,181,165,95)(71,182,166,96)(72,170,167,97)(73,171,168,98)(74,172,169,99)(75,173,157,100)(76,174,158,101)(77,175,159,102)(78,176,160,103), (14,184)(15,185)(16,186)(17,187)(18,188)(19,189)(20,190)(21,191)(22,192)(23,193)(24,194)(25,195)(26,183)(40,132)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,131)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,105)(65,106)(66,161)(67,162)(68,163)(69,164)(70,165)(71,166)(72,167)(73,168)(74,169)(75,157)(76,158)(77,159)(78,160) );
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)], [(1,179,127,90,155,93,31,204),(2,180,128,91,156,94,32,205),(3,181,129,79,144,95,33,206),(4,182,130,80,145,96,34,207),(5,170,118,81,146,97,35,208),(6,171,119,82,147,98,36,196),(7,172,120,83,148,99,37,197),(8,173,121,84,149,100,38,198),(9,174,122,85,150,101,39,199),(10,175,123,86,151,102,27,200),(11,176,124,87,152,103,28,201),(12,177,125,88,153,104,29,202),(13,178,126,89,154,92,30,203),(14,73,133,106,184,168,41,65),(15,74,134,107,185,169,42,53),(16,75,135,108,186,157,43,54),(17,76,136,109,187,158,44,55),(18,77,137,110,188,159,45,56),(19,78,138,111,189,160,46,57),(20,66,139,112,190,161,47,58),(21,67,140,113,191,162,48,59),(22,68,141,114,192,163,49,60),(23,69,142,115,193,164,50,61),(24,70,143,116,194,165,51,62),(25,71,131,117,195,166,52,63),(26,72,132,105,183,167,40,64)], [(1,192,155,22),(2,193,156,23),(3,194,144,24),(4,195,145,25),(5,183,146,26),(6,184,147,14),(7,185,148,15),(8,186,149,16),(9,187,150,17),(10,188,151,18),(11,189,152,19),(12,190,153,20),(13,191,154,21),(27,137,123,45),(28,138,124,46),(29,139,125,47),(30,140,126,48),(31,141,127,49),(32,142,128,50),(33,143,129,51),(34,131,130,52),(35,132,118,40),(36,133,119,41),(37,134,120,42),(38,135,121,43),(39,136,122,44),(53,197,107,83),(54,198,108,84),(55,199,109,85),(56,200,110,86),(57,201,111,87),(58,202,112,88),(59,203,113,89),(60,204,114,90),(61,205,115,91),(62,206,116,79),(63,207,117,80),(64,208,105,81),(65,196,106,82),(66,177,161,104),(67,178,162,92),(68,179,163,93),(69,180,164,94),(70,181,165,95),(71,182,166,96),(72,170,167,97),(73,171,168,98),(74,172,169,99),(75,173,157,100),(76,174,158,101),(77,175,159,102),(78,176,160,103)], [(14,184),(15,185),(16,186),(17,187),(18,188),(19,189),(20,190),(21,191),(22,192),(23,193),(24,194),(25,195),(26,183),(40,132),(41,133),(42,134),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,141),(50,142),(51,143),(52,131),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(61,115),(62,116),(63,117),(64,105),(65,106),(66,161),(67,162),(68,163),(69,164),(70,165),(71,166),(72,167),(73,168),(74,169),(75,157),(76,158),(77,159),(78,160)]])
260 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AV | 52A | ··· | 52X | 52Y | ··· | 52BH | 104A | ··· | 104AV | 104AW | ··· | 104DP |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
260 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C13 | C26 | C26 | C26 | C52 | C52 | C8○D4 | C13×C8○D4 |
| kernel | C13×C8○D4 | C2×C104 | C13×M4(2) | C13×C4○D4 | D4×C13 | Q8×C13 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C13 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 12 | 36 | 36 | 12 | 72 | 24 | 4 | 48 |
Matrix representation of C13×C8○D4 ►in GL2(𝔽313) generated by
| 113 | 0 |
| 0 | 113 |
| 5 | 0 |
| 0 | 5 |
| 265 | 2 |
| 256 | 48 |
| 1 | 0 |
| 48 | 312 |
G:=sub<GL(2,GF(313))| [113,0,0,113],[5,0,0,5],[265,256,2,48],[1,48,0,312] >;
C13×C8○D4 in GAP, Magma, Sage, TeX
C_{13}\times C_8\circ D_4 % in TeX
G:=Group("C13xC8oD4"); // GroupNames label
G:=SmallGroup(416,192);
// by ID
G=gap.SmallGroup(416,192);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1916,88]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations