direct product, p-group, abelian, monomial
Aliases: C2×C4×C8, SmallGroup(64,83)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2×C4×C8 |
| C1 — C2×C4×C8 |
| C1 — C2×C4×C8 |
Generators and relations for C2×C4×C8
G = < a,b,c | a2=b4=c8=1, ab=ba, ac=ca, bc=cb >
Subgroups: 81, all normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C4×C8, C2×C42, C22×C8, C2×C4×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C2×C4×C8
►Regular action on 64 points
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)
(1 47 19 13)(2 48 20 14)(3 41 21 15)(4 42 22 16)(5 43 23 9)(6 44 24 10)(7 45 17 11)(8 46 18 12)(25 36 58 52)(26 37 59 53)(27 38 60 54)(28 39 61 55)(29 40 62 56)(30 33 63 49)(31 34 64 50)(32 35 57 51)
(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)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58), (1,47,19,13)(2,48,20,14)(3,41,21,15)(4,42,22,16)(5,43,23,9)(6,44,24,10)(7,45,17,11)(8,46,18,12)(25,36,58,52)(26,37,59,53)(27,38,60,54)(28,39,61,55)(29,40,62,56)(30,33,63,49)(31,34,64,50)(32,35,57,51), (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( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58), (1,47,19,13)(2,48,20,14)(3,41,21,15)(4,42,22,16)(5,43,23,9)(6,44,24,10)(7,45,17,11)(8,46,18,12)(25,36,58,52)(26,37,59,53)(27,38,60,54)(28,39,61,55)(29,40,62,56)(30,33,63,49)(31,34,64,50)(32,35,57,51), (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([[(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,57),(48,58)], [(1,47,19,13),(2,48,20,14),(3,41,21,15),(4,42,22,16),(5,43,23,9),(6,44,24,10),(7,45,17,11),(8,46,18,12),(25,36,58,52),(26,37,59,53),(27,38,60,54),(28,39,61,55),(29,40,62,56),(30,33,63,49),(31,34,64,50),(32,35,57,51)], [(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)]])
C2×C4×C8 is a maximal subgroup of
C2.C82 C42.385D4 M4(2)⋊C8 C42.46Q8 C22.7M5(2) C42.7C8 C82⋊C2 C8⋊9M4(2) C23.27C42 C42.455D4 C42.315D4 C42.316D4 C42.305D4 C42.42Q8 C8⋊8M4(2) C8⋊7M4(2) C42.43Q8 C2.C43 C42⋊4C8 (C4×C8)⋊12C4 C42.379D4 C23.17C42 Q8.C42 C42.45Q8 C4⋊C8⋊13C4 C4⋊C8⋊14C4 C8.14C42 C42.55Q8 C42.56Q8 C42.322D4 C42.58Q8 C42.59Q8 C42.60Q8 C42.324D4 C4⋊C4⋊3C8 C22⋊C4⋊4C8 C2.(C8⋊8D4) C2.(C8⋊7D4) C42.428D4 C42.61Q8 C8⋊7(C4⋊C4) C8⋊5(C4⋊C4) C42.62Q8 C42.325D4 C42.431D4 C42.432D4 C42.433D4 (C2×C4)⋊9SD16 (C2×C4)⋊6Q16 (C2×C4)⋊6D8 C42.326D4 C42.327D4 C42.436D4 C42.437D4 C4⋊M5(2) C42.13C8 C42.6C8 C42.260C23 C42.681C23 C42.286C23 C42.290C23 C42.291C23 C42.355D4 C42.360D4 C42.364D4 C42.365D4 C42.308D4 C42.366D4 C42.367D4
C2×C4×C8 is a maximal quotient of
C82⋊C2 C42⋊4C8 C16○2M5(2)
64 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 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 | 1 | 1 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 |
| kernel | C2×C4×C8 | C4×C8 | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 32 |
Matrix representation of C2×C4×C8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 16 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 4 |
| 2 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 9 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,13,0,0,0,4],[2,0,0,0,16,0,0,0,9] >;
C2×C4×C8 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_8
% in TeX
G:=Group("C2xC4xC8"); // GroupNames label
G:=SmallGroup(64,83);
// by ID
G=gap.SmallGroup(64,83);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,117]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^8=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations