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## G = C2×C42⋊3C4order 128 = 27

### Direct product of C2 and C42⋊3C4

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C42⋊3C4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C22×D4 — C2×C4.4D4 — C2×C42⋊3C4
 Lower central C1 — C2 — C22 — C2×C4 — C2×C42⋊3C4
 Upper central C1 — C22 — C23 — C22×D4 — C2×C42⋊3C4
 Jennings C1 — C2 — C22 — C2×D4 — C2×C42⋊3C4

Generators and relations for C2×C423C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >

Subgroups: 388 in 142 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×10], C22 [×3], C22 [×16], C2×C4 [×2], C2×C4 [×18], D4 [×4], Q8 [×4], C23, C23 [×4], C23 [×8], C42 [×2], C42, C22⋊C4 [×14], C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×3], C24 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×4], C4.4D4 [×2], C22×D4, C22×Q8, C423C4 [×4], C2×C23⋊C4 [×2], C2×C4.4D4, C2×C423C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C423C4 [×2], C2×C23⋊C4, C2×C423C4

Character table of C2×C423C4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ3 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ6 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 linear of order 2 ρ7 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 linear of order 2 ρ9 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 i i 1 -i -i i -1 i -i -i linear of order 4 ρ10 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -i i -1 -i i i -1 -i -i i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 i -i -1 i -i -i -1 i i -i linear of order 4 ρ12 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 -i -i 1 i i -i -1 -i i i linear of order 4 ρ13 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -i i -1 i i -i 1 i -i -i linear of order 4 ρ14 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 1 -1 i i 1 i -i -i 1 -i -i i linear of order 4 ρ15 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 1 -1 -i -i 1 -i i i 1 i i -i linear of order 4 ρ16 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 i -i -1 -i -i i 1 -i i i linear of order 4 ρ17 2 2 2 2 2 2 -2 2 2 -2 0 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 -2 2 -2 2 2 -2 2 -2 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 2 -2 2 -2 2 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 2 -2 -2 2 0 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 -4 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 -4 4 -4 0 0 0 0 0 0 2i 0 -2i -2i 0 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4 ρ24 4 4 -4 -4 0 0 0 0 0 0 -2i 0 -2i 2i 0 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4 ρ25 4 4 -4 -4 0 0 0 0 0 0 2i 0 2i -2i 0 -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4 ρ26 4 -4 4 -4 0 0 0 0 0 0 -2i 0 2i 2i 0 -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4

Smallest permutation representation of C2×C423C4
On 32 points
Generators in S32
(1 7)(2 8)(3 5)(4 6)(9 26)(10 27)(11 28)(12 25)(13 22)(14 23)(15 24)(16 21)(17 31)(18 32)(19 29)(20 30)
(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 27 30 14)(2 28 31 15)(3 25 32 16)(4 26 29 13)(5 12 18 21)(6 9 19 22)(7 10 20 23)(8 11 17 24)
(1 9 14 8)(2 7 26 23)(3 24 16 19)(4 18 28 12)(5 15 21 29)(6 32 11 25)(10 31 20 13)(17 30 22 27)

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

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

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

Matrix representation of C2×C423C4 in GL6(𝔽5)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 4 0 0 0 0 4 0 0 0 0 0 0 0 2 3 2 2 0 0 0 3 0 0 0 0 3 2 3 0 0 0 3 2 0 3
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 3 0 0 0 0 0 4 1 0 0 1 0 1 0 0 0 1 4 1 0
,
 0 2 0 0 0 0 3 0 0 0 0 0 0 0 2 2 2 2 0 0 2 0 2 2 0 0 0 3 0 3 0 0 3 0 0 3

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,0,3,3,0,0,3,3,2,2,0,0,2,0,3,0,0,0,2,0,0,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,1,1,0,0,0,0,0,4,0,0,3,4,1,1,0,0,0,1,0,0],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,2,2,0,3,0,0,2,0,3,0,0,0,2,2,0,0,0,0,2,2,3,3] >;

C2×C423C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_3C_4
% in TeX

G:=Group("C2xC4^2:3C4");
// GroupNames label

G:=SmallGroup(128,857);
// by ID

G=gap.SmallGroup(128,857);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,1018,248,1971,375,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations

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