direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4×D4, C42⋊16C22, C22.8C24, C23.29C23, C24.29C22, C4○(C4×D4), C4⋊1(C22×C4), (C23×C4)⋊3C2, (C2×C42)⋊7C2, C23⋊4(C2×C4), C4⋊C4⋊19C22, C2.4(C23×C4), C2.3(C22×D4), C22⋊1(C22×C4), C22.59(C2×D4), C22⋊C4⋊17C22, (C2×C4).158C23, (C22×C4)⋊16C22, (C22×D4).13C2, (C2×D4).75C22, C22.27(C4○D4), C4○2(C2×C4⋊C4), (C2×C4)○(C4×D4), (C2×C4)⋊7(C2×C4), C4⋊C4○(C22×C4), (C2×C4⋊C4)⋊25C2, (C2×C4)○3(C4⋊C4), C4○2(C2×C22⋊C4), C2.2(C2×C4○D4), (C2×C4)○(C22×D4), (C22×C4)○(C2×D4), C22⋊C4○(C22×C4), (C2×C4)○3(C22⋊C4), (C2×C22⋊C4)⋊16C2, (C22×C4)○(C22×D4), (C2×C4)○2(C2×C4⋊C4), (C22×C4)○(C2×C4⋊C4), (C2×C4)○2(C2×C22⋊C4), (C22×C4)○(C2×C22⋊C4), SmallGroup(64,196)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4×D4
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 289 in 213 conjugacy classes, 137 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C2×C4×D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4
►On 32 points
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)
(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)
(1 28 23 13)(2 25 24 14)(3 26 21 15)(4 27 22 16)(5 17 10 30)(6 18 11 31)(7 19 12 32)(8 20 9 29)
(1 8)(2 5)(3 6)(4 7)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,20)(14,17)(15,18)(16,19)(25,30)(26,31)(27,32)(28,29), (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), (1,28,23,13)(2,25,24,14)(3,26,21,15)(4,27,22,16)(5,17,10,30)(6,18,11,31)(7,19,12,32)(8,20,9,29), (1,8)(2,5)(3,6)(4,7)(9,23)(10,24)(11,21)(12,22)(13,20)(14,17)(15,18)(16,19)(25,30)(26,31)(27,32)(28,29)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,20)(14,17)(15,18)(16,19)(25,30)(26,31)(27,32)(28,29), (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), (1,28,23,13)(2,25,24,14)(3,26,21,15)(4,27,22,16)(5,17,10,30)(6,18,11,31)(7,19,12,32)(8,20,9,29), (1,8)(2,5)(3,6)(4,7)(9,23)(10,24)(11,21)(12,22)(13,20)(14,17)(15,18)(16,19)(25,30)(26,31)(27,32)(28,29) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,24),(6,21),(7,22),(8,23),(13,20),(14,17),(15,18),(16,19),(25,30),(26,31),(27,32),(28,29)], [(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)], [(1,28,23,13),(2,25,24,14),(3,26,21,15),(4,27,22,16),(5,17,10,30),(6,18,11,31),(7,19,12,32),(8,20,9,29)], [(1,8),(2,5),(3,6),(4,7),(9,23),(10,24),(11,21),(12,22),(13,20),(14,17),(15,18),(16,19),(25,30),(26,31),(27,32),(28,29)]])
C2×C4×D4 is a maximal subgroup of
C23.8M4(2) C42.393D4 C23⋊M4(2) C42.43D4 C42.398D4 D4⋊M4(2) D4⋊5M4(2) D4⋊C42 C24.167C23 C42.96D4 C42.98D4 C42.100D4 C2.(C4×D8) D4⋊(C4⋊C4) C23.22M4(2) C23⋊2M4(2) (C2×SD16)⋊14C4 (C2×C4)⋊9D8 C42.325D4 C42.109D4 D4⋊4C42 C42⋊42D4 C43⋊9C2 C24.547C23 C23.201C24 C23.203C24 C24.195C23 C42⋊13D4 C24.198C23 C42.160D4 C42⋊14D4 C24.549C23 C23.231C24 C23.234C24 C23.235C24 C23.236C24 C24.212C23 C23.240C24 C23.241C24 C24.215C23 C24.217C23 C24.218C23 C24.219C23 C24.220C23 C23.288C24 C42⋊15D4 C23.295C24 C42⋊16D4 C42.163D4 C24.244C23 C23.308C24 C23.309C24 C24.249C23 C23.315C24 C23.316C24 C24.252C23 C23.318C24 C24.563C23 C24.254C23 C23.322C24 C23.324C24 C24.258C23 C23.327C24 C23.328C24 C24.269C23 C23.344C24 C23.345C24 C24.271C23 C23.349C24 C23.350C24 C23.352C24 C23.354C24 C24.276C23 C23.356C24 C24.278C23 C23.359C24 C23.360C24 C24.282C23 C24.283C23 C23.364C24 C24.286C23 C23.367C24 C23.368C24 C23.385C24 C24.299C23 C24.300C23 C23.434C24 C42⋊17D4 C42.165D4 C42⋊18D4 C42.166D4 C42⋊19D4 C42⋊20D4 C42.167D4 C23.443C24 C42⋊21D4 C42.170D4 C42.172D4 C42.173D4 C24.583C23 C42.175D4 C23.479C24 C42.178D4 C42⋊22D4 C23.500C24 C23.502C24 C42⋊24D4 C23.530C24 C42⋊29D4 C42.190D4 C23.535C24 C42⋊30D4 C24.374C23 C42.691C23 C23⋊3M4(2) D4⋊7M4(2) C42.693C23 C42.211D4 C42.219D4 C42.221D4 C42.222D4 C42.225D4 C42.227D4 C42.228D4 C42.232D4 C22.48C25 C22.49C25 C22.64C25 C22.70C25 C4⋊2+ 1+4 C22.90C25 C22.94C25 C22.95C25 C22.102C25 C22.108C25 C23.144C24
C2×C4×D4 is a maximal quotient of
C42⋊42D4 C43⋊9C2 C23.203C24 C24.195C23 C42.159D4 C42⋊13D4 C24.198C23 C42.160D4 C42.161D4 C42⋊14D4 C23.224C24 C23.226C24 C23.234C24 C23.236C24 C23.240C24 C23.241C24 C24.558C23 C24.215C23 C23.244C24 C24.217C23 C24.218C23 C23.247C24 C24.219C23 C24.220C23 C42.264C23 C42.265C23 C42.681C23 C42.266C23 M4(2)⋊22D4 M4(2)⋊23D4 C42.383D4 C42.275C23 C42.276C23 C42.277C23 C42.278C23 C42.279C23 C42.280C23 C42.281C23 C42.283C23 M4(2).51D4 M4(2)○D8
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 |
| kernel | C2×C4×D4 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C4×D4 | C23×C4 | C22×D4 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 16 | 4 | 4 |
Matrix representation of C2×C4×D4 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,4,0,0,0,0,0,1,0,0,4,0],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4] >;
C2×C4×D4 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_4
% in TeX
G:=Group("C2xC4xD4"); // GroupNames label
G:=SmallGroup(64,196);
// by ID
G=gap.SmallGroup(64,196);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,158]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations