direct product, p-group, abelian, monomial
Aliases: C23×C8, SmallGroup(64,246)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C23×C8 |
| C1 — C23×C8 |
| C1 — C23×C8 |
Generators and relations for C23×C8
G = < a,b,c,d | a2=b2=c2=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 169, all normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, C23×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, C23×C8
►Regular action on 64 points
Generators in S64
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(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)
G:=sub<Sym(64)| (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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)>;
G:=Group( (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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) );
G=PermutationGroup([[(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,45),(18,46),(19,47),(20,48),(21,41),(22,42),(23,43),(24,44),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42)], [(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)]])
C23×C8 is a maximal subgroup of
C23.29C42 C24.132D4 C23.36C42 C24.133D4 C23.22D8 C24.19Q8 C23.21M4(2) C23.22M4(2) C24.135D4 C23.23D8 C24.5C8 C42.264C23 C24.144D4
C23×C8 is a maximal quotient of
C42.691C23 C42.695C23 C42.697C23 Q8○M5(2)
64 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | C8 |
| kernel | C23×C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 |
| # reps | 1 | 14 | 1 | 14 | 2 | 32 |
Matrix representation of C23×C8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 8 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[8,0,0,0,0,15,0,0,0,0,4,0,0,0,0,13] >;
C23×C8 in GAP, Magma, Sage, TeX
C_2^3\times C_8
% in TeX
G:=Group("C2^3xC8"); // GroupNames label
G:=SmallGroup(64,246);
// by ID
G=gap.SmallGroup(64,246);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,96,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations