direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22⋊C8, C23⋊2C8, C24.3C4, C22.9M4(2), C4○(C22⋊C8), (C22×C8)⋊1C2, C22⋊2(C2×C8), C4.66(C2×D4), (C2×C8)⋊10C22, (C2×C4).144D4, C2.1(C22×C8), (C23×C4).5C2, (C22×C4).11C4, C23.28(C2×C4), C2.3(C2×M4(2)), C4.31(C22⋊C4), (C2×C4).144C23, (C22×C4).91C22, C22.20(C22×C4), C22.28(C22⋊C4), (C2×C4)○(C22⋊C8), (C2×C4).55(C2×C4), C2.3(C2×C22⋊C4), SmallGroup(64,87)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22⋊C8
G = < a,b,c,d | a2=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 145 in 101 conjugacy classes, 57 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C22×C8, C23×C4, C2×C22⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8
►On 32 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 5)(2 32)(3 7)(4 26)(6 28)(8 30)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)(25 29)(27 31)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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)
G:=sub<Sym(32)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (1,5)(2,32)(3,7)(4,26)(6,28)(8,30)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23)(25,29)(27,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (1,5)(2,32)(3,7)(4,26)(6,28)(8,30)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23)(25,29)(27,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29)], [(1,5),(2,32),(3,7),(4,26),(6,28),(8,30),(9,18),(10,14),(11,20),(12,16),(13,22),(15,24),(17,21),(19,23),(25,29),(27,31)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(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)]])
C2×C22⋊C8 is a maximal subgroup of
C23.19C42 C23.21C42 C23.8D8 C24.2Q8 C23.30D8 C24.3Q8 C23⋊C16 C23.8M4(2) C23⋊C8⋊C2 C24.(C2×C4) C24.45(C2×C4) C24.53D4 C24.59D4 C24.60D4 C24.61D4 C42.378D4 C42.379D4 C8×C22⋊C4 C23.36C42 C23.17C42 C24⋊3C8 C24.51(C2×C4) C23.35D8 C24.155D4 C24.65D4 C42.425D4 C42.95D4 C23.32M4(2) C24.53(C2×C4) C23.36D8 C24.157D4 C24.69D4 C23.21M4(2) (C2×C8).195D4 C23.37D8 C24.159D4 C24.71D4 C24.10Q8 C23.22M4(2) C23⋊2M4(2) C24.160D4 C24.73D4 C23.38D8 C24.74D4 C22⋊C4⋊4C8 C23.9M4(2) C42.325D4 C42.109D4 C23⋊2D8 C23⋊3SD16 C23⋊2Q16 C24.83D4 C24.84D4 C24.85D4 C24.86D4 C23.12D8 C24.88D4 C24.89D4 D4○(C22⋊C8) C42.262C23 D4×C2×C8 M4(2)⋊22D4 C42.691C23 C23⋊3M4(2) D4⋊7M4(2) C42.297C23 C42.298C23 C24.103D4 C24.115D4 C23⋊3D8 C23⋊4SD16 C24.121D4 C23⋊3Q16 C24.123D4 C24.124D4 D10.11M4(2)
C2×C22⋊C8 is a maximal quotient of
C42.371D4 C23.8M4(2) C42.393D4 C42.394D4 C42.455D4 C42.397D4 C42.398D4 C42.399D4 C24⋊3C8 C42.425D4 C23.32M4(2) C23.22M4(2) C42.325D4 C42.327D4 C24.5C8 (C2×D4).5C8 M5(2).19C22 M5(2)⋊12C22 D10.11M4(2)
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) |
| kernel | C2×C22⋊C8 | C22⋊C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 16 | 4 | 4 |
Matrix representation of C2×C22⋊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 | 1 | 0 |
| 0 | 0 | 8 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 10 | 8 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,1,8,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[9,0,0,0,0,16,0,0,0,0,9,10,0,0,2,8] >;
C2×C22⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes C_8
% in TeX
G:=Group("C2xC2^2:C8"); // GroupNames label
G:=SmallGroup(64,87);
// by ID
G=gap.SmallGroup(64,87);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,88]);
// 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,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations