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## G = (C2×C8).6D4order 128 = 27

### 6th non-split extension by C2×C8 of D4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).6D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — C2×C4.10D4 — (C2×C8).6D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).6D4
 Upper central C1 — C2 — C22×C4 — (C2×C8).6D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).6D4

Generators and relations for (C2×C8).6D4
G = < a,b,c,d | a2=b8=1, c4=d2=b4, cbc-1=ab=ba, cac-1=ab4, ad=da, dbd-1=ab-1, dcd-1=ab2c3 >

Subgroups: 184 in 92 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×4], C22 [×3], C22, C8 [×5], C2×C4 [×6], C2×C4 [×6], Q8 [×6], C23, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×3], M4(2) [×8], C22×C4, C22×C4, C2×Q8 [×2], C2×Q8 [×5], C4.10D4 [×2], Q8⋊C4 [×4], C8.C4 [×2], C42⋊C2 [×2], C2×M4(2) [×4], C22×Q8, C4.9C42, M4(2)⋊4C4 [×2], C2×C4.10D4, C23.38D4 [×2], M4(2).C4, (C2×C8).6D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C23.11D4, (C2×C8).6D4

Character table of (C2×C8).6D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 2 2 2 2 8 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 2 0 -2 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 -2 2 2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 2 -2 -2 2 2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 2 2 -2 2 -2 0 2i 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 complex lifted from C4○D4 ρ15 2 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 complex lifted from C4○D4 ρ16 2 2 -2 -2 2 2 -2 2 -2 0 -2i 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ17 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 2i 0 0 0 0 0 0 -2i complex lifted from C4○D4 ρ19 2 2 2 -2 -2 2 2 -2 -2 0 0 2i 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 -2i 0 0 0 0 0 0 2i complex lifted from C4○D4 ρ22 2 2 2 -2 -2 2 2 -2 -2 0 0 -2i 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C2×C8).6D4
On 32 points
Generators in S32
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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 31 26 6 5 27 30 2)(3 25 28 8 7 29 32 4)(9 14 18 19 13 10 22 23)(11 16 20 21 15 12 24 17)
(1 9 5 13)(2 19 6 23)(3 15 7 11)(4 17 8 21)(10 31 14 27)(12 29 16 25)(18 26 22 30)(20 32 24 28)

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

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

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

Matrix representation of (C2×C8).6D4 in GL8(𝔽17)

 16 0 0 2 0 0 0 0 16 0 1 1 0 0 0 0 16 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 1 0 0 0 1 0 0 0 1 0 0 16 0 0 0 0 1 0 16 0 0 16 0 0 1 1 0 0 0
,
 10 0 0 0 0 7 0 7 0 0 0 0 5 12 5 12 5 0 0 0 0 7 0 0 5 0 0 0 12 12 5 12 5 0 12 0 0 12 0 12 5 12 0 5 0 12 0 12 0 0 5 0 0 12 0 12 10 12 0 12 0 12 0 12
,
 7 0 0 0 10 0 10 0 7 0 0 0 5 5 5 5 12 0 0 0 0 0 10 0 12 0 0 0 5 5 5 12 12 0 12 0 5 0 5 0 12 12 0 5 5 0 5 0 0 0 12 0 5 0 5 0 7 5 0 5 5 0 5 0
,
 10 0 7 0 0 0 0 0 10 12 12 12 0 0 0 0 5 0 7 0 0 0 0 0 5 12 12 5 0 0 0 0 10 0 12 0 12 0 12 0 0 0 12 0 0 5 0 5 5 0 12 0 12 0 5 0 5 0 12 0 0 5 0 12

G:=sub<GL(8,GF(17))| [16,16,16,0,16,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0],[10,0,5,5,5,5,0,10,0,0,0,0,0,12,0,12,0,0,0,0,12,0,5,0,0,0,0,0,0,5,0,12,0,5,0,12,0,0,0,0,7,12,7,12,12,12,12,12,0,5,0,5,0,0,0,0,7,12,0,12,12,12,12,12],[7,7,12,12,12,12,0,7,0,0,0,0,0,12,0,5,0,0,0,0,12,0,12,0,0,0,0,0,0,5,0,5,10,5,0,5,5,5,5,5,0,5,0,5,0,0,0,0,10,5,10,5,5,5,5,5,0,5,0,12,0,0,0,0],[10,10,5,5,10,0,5,5,0,12,0,12,0,0,0,0,7,12,7,12,12,12,12,12,0,12,0,5,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,5,0,5,0,0,0,0,12,0,5,0,0,0,0,0,0,5,0,12] >;

(C2×C8).6D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._6D_4
% in TeX

G:=Group("(C2xC8).6D4");
// GroupNames label

G:=SmallGroup(128,814);
// by ID

G=gap.SmallGroup(128,814);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,456,422,387,58,1018,248,1411,718,172,4037,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=d^2=b^4,c*b*c^-1=a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,d*b*d^-1=a*b^-1,d*c*d^-1=a*b^2*c^3>;
// generators/relations

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