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G = C2×C4×D4order 64 = 26

Direct product of C2×C4 and D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C4×D4, C4216C22, C22.8C24, C23.29C23, C24.29C22, C4(C4×D4), C41(C22×C4), (C23×C4)⋊3C2, (C2×C42)⋊7C2, C234(C2×C4), C4⋊C419C22, C2.4(C23×C4), C2.3(C22×D4), C221(C22×C4), C22.59(C2×D4), C22⋊C417C22, (C2×C4).158C23, (C22×C4)⋊16C22, (C22×D4).13C2, (C2×D4).75C22, C22.27(C4○D4), C42(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), C42(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

C1C2 — C2×C4×D4
C1C2C22C23C22×C4C23×C4 — C2×C4×D4
C1C2 — C2×C4×D4
C1C22×C4 — C2×C4×D4
C1C22 — C2×C4×D4

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 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×6], C22, C22 [×14], C22 [×24], C2×C4 [×18], C2×C4 [×22], D4 [×16], C23, C23 [×12], C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×12], C24 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C2×C4×D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4

Smallest permutation representation of C2×C4×D4
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)
(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 26 23 13)(2 27 24 14)(3 28 21 15)(4 25 22 16)(5 17 12 32)(6 18 9 29)(7 19 10 30)(8 20 11 31)
(1 6)(2 7)(3 8)(4 5)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)

G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (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), (1,26,23,13)(2,27,24,14)(3,28,21,15)(4,25,22,16)(5,17,12,32)(6,18,9,29)(7,19,10,30)(8,20,11,31), (1,6)(2,7)(3,8)(4,5)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31)>;

G:=Group( (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (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), (1,26,23,13)(2,27,24,14)(3,28,21,15)(4,25,22,16)(5,17,12,32)(6,18,9,29)(7,19,10,30)(8,20,11,31), (1,6)(2,7)(3,8)(4,5)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31) );

G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,18),(14,19),(15,20),(16,17),(25,32),(26,29),(27,30),(28,31)], [(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)], [(1,26,23,13),(2,27,24,14),(3,28,21,15),(4,25,22,16),(5,17,12,32),(6,18,9,29),(7,19,10,30),(8,20,11,31)], [(1,6),(2,7),(3,8),(4,5),(9,23),(10,24),(11,21),(12,22),(13,18),(14,19),(15,20),(16,17),(25,32),(26,29),(27,30),(28,31)])

C2×C4×D4 is a maximal subgroup of
C23.8M4(2)  C42.393D4  C23⋊M4(2)  C42.43D4  C42.398D4  D4⋊M4(2)  D45M4(2)  D4⋊C42  C24.167C23  C42.96D4  C42.98D4  C42.100D4  C2.(C4×D8)  D4⋊(C4⋊C4)  C23.22M4(2)  C232M4(2)  (C2×SD16)⋊14C4  (C2×C4)⋊9D8  C42.325D4  C42.109D4  D44C42  C4242D4  C439C2  C24.547C23  C23.201C24  C23.203C24  C24.195C23  C4213D4  C24.198C23  C42.160D4  C4214D4  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  C4215D4  C23.295C24  C4216D4  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  C4217D4  C42.165D4  C4218D4  C42.166D4  C4219D4  C4220D4  C42.167D4  C23.443C24  C4221D4  C42.170D4  C42.172D4  C42.173D4  C24.583C23  C42.175D4  C23.479C24  C42.178D4  C4222D4  C23.500C24  C23.502C24  C4224D4  C23.530C24  C4229D4  C42.190D4  C23.535C24  C4230D4  C24.374C23  C42.691C23  C233M4(2)  D47M4(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
C4242D4  C439C2  C23.203C24  C24.195C23  C42.159D4  C4213D4  C24.198C23  C42.160D4  C42.161D4  C4214D4  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···2G2H···2O4A···4H4I···4X
order12···22···24···44···4
size11···12···21···12···2

40 irreducible representations

dim1111111122
type++++++++
imageC1C2C2C2C2C2C2C4D4C4○D4
kernelC2×C4×D4C2×C42C2×C22⋊C4C2×C4⋊C4C4×D4C23×C4C22×D4C2×D4C2×C4C22
# reps11218211644

Matrix representation of C2×C4×D4 in GL4(𝔽5) generated by

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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

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