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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 — C2×C4 — C2×Q8 — C22×Q8 — C2×C4⋊Q8 — C2×C42.3C4
 Lower central C1 — C2 — C22 — C2×C4 — C2×C42.3C4
 Upper central C1 — C22 — C23 — C22×Q8 — C2×C42.3C4
 Jennings C1 — C2 — C22 — C2×Q8 — C2×C42.3C4

Generators and relations for C2×C42.3C4
G = < a,b,c,d | a2=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >

Subgroups: 244 in 120 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×12], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×16], Q8 [×8], C23, C42 [×2], C42, C4⋊C4 [×8], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×4], C2×Q8 [×5], C4.10D4 [×4], C4.10D4 [×2], C2×C42, C2×C4⋊C4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8 [×2], C42.3C4 [×4], C2×C4.10D4 [×2], C2×C4⋊Q8, C2×C42.3C4
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, C42.3C4 [×2], C2×C23⋊C4, C2×C42.3C4

Character table of C2×C42.3C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H 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 1 1 i -i i -i i i -i -i linear of order 4 ρ10 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -i -i i -i -i i i i linear of order 4 ρ11 1 1 1 1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 -i -i -i i i i i -i linear of order 4 ρ12 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 i -i -i i -i i -i i linear of order 4 ρ13 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 -i i -i i -i -i i i linear of order 4 ρ14 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 i i -i i i -i -i -i linear of order 4 ρ15 1 1 1 1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 i i i -i -i -i -i i linear of order 4 ρ16 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -i i i -i i -i i -i linear of order 4 ρ17 2 2 2 2 2 2 0 -2 0 0 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 2 -2 0 2 0 0 2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 0 -2 0 0 -2 -2 2 2 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 0 2 0 0 -2 -2 2 -2 0 -2 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 2 0 2 -2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ24 4 4 -4 -4 0 0 -2 0 -2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ25 4 -4 -4 4 0 0 2 0 -2 -2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ26 4 -4 -4 4 0 0 -2 0 2 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2

Smallest permutation representation of C2×C42.3C4
On 32 points
Generators in S32
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(2 28 6 32)(4 26 8 30)(9 24 13 20)(11 22 15 18)
(1 27 5 31)(2 28 6 32)(3 25 7 29)(4 26 8 30)(9 24 13 20)(10 21 14 17)(11 22 15 18)(12 19 16 23)
(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)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (2,28,6,32)(4,26,8,30)(9,24,13,20)(11,22,15,18), (1,27,5,31)(2,28,6,32)(3,25,7,29)(4,26,8,30)(9,24,13,20)(10,21,14,17)(11,22,15,18)(12,19,16,23), (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( (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (2,28,6,32)(4,26,8,30)(9,24,13,20)(11,22,15,18), (1,27,5,31)(2,28,6,32)(3,25,7,29)(4,26,8,30)(9,24,13,20)(10,21,14,17)(11,22,15,18)(12,19,16,23), (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([(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(2,28,6,32),(4,26,8,30),(9,24,13,20),(11,22,15,18)], [(1,27,5,31),(2,28,6,32),(3,25,7,29),(4,26,8,30),(9,24,13,20),(10,21,14,17),(11,22,15,18),(12,19,16,23)], [(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)])

Matrix representation of C2×C42.3C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 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
,
 1 0 0 0 0 0 15 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 16 0 0 0 0 16 7
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 7 1 0 0 0 0 1 10 0 0 0 0 0 0 10 16 0 0 0 0 16 7
,
 1 1 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 16 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,1,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[1,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,16,0,0] >;

C2×C42.3C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2._3C_4
% in TeX

G:=Group("C2xC4^2.3C4");
// GroupNames label

G:=SmallGroup(128,863);
// by ID

G=gap.SmallGroup(128,863);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,1018,248,1971,375,172,4037]);
// Polycyclic

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

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