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## G = (C2×C4).D8order 128 = 27

### 4th non-split extension by C2×C4 of D8 acting via D8/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4).D8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.81C23 — (C2×C4).D8
 Lower central C1 — C2 — C23 — C22×C4 — (C2×C4).D8
 Upper central C1 — C22 — C23 — C2×C4⋊C4 — (C2×C4).D8
 Jennings C1 — C22 — C23 — C2×C4⋊C4 — (C2×C4).D8

Generators and relations for (C2×C4).D8
G = < a,b,c,d | a2=b4=c8=1, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=ab-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 156 in 62 conjugacy classes, 20 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×7], C22 [×3], C22 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×13], C23, C4⋊C4 [×4], C2×C8 [×2], M4(2) [×3], C22×C4 [×3], C22×C4 [×2], C2.C42, C2.C42, C22⋊C8, C2×C4⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C22.M4(2), C22.C42, C23.81C23, (C2×C4).D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, C23⋊C4, D4⋊C4, C4≀C2, C22.SD16, C23.D4, C42.3C4, (C2×C4).D8

Character table of (C2×C4).D8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 4 4 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 -i i i i -i -i -i i linear of order 4 ρ6 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 ρ7 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 ρ8 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 ρ9 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 ρ10 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 ρ11 2 -2 2 -2 2 -2 2 0 -2 0 0 0 0 0 0 √2 0 0 -√2 0 0 -√2 √2 orthogonal lifted from D8 ρ12 2 -2 2 -2 2 -2 2 0 -2 0 0 0 0 0 0 -√2 0 0 √2 0 0 √2 -√2 orthogonal lifted from D8 ρ13 2 -2 2 -2 -2 2 0 -2i 0 2i 0 0 0 0 0 0 -1+i 1-i 0 -1-i 1+i 0 0 complex lifted from C4≀C2 ρ14 2 -2 2 -2 -2 2 0 2i 0 -2i 0 0 0 0 0 0 -1-i 1+i 0 -1+i 1-i 0 0 complex lifted from C4≀C2 ρ15 2 -2 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 √-2 0 0 √-2 0 0 -√-2 -√-2 complex lifted from SD16 ρ16 2 -2 2 -2 -2 2 0 2i 0 -2i 0 0 0 0 0 0 1+i -1-i 0 1-i -1+i 0 0 complex lifted from C4≀C2 ρ17 2 -2 2 -2 -2 2 0 -2i 0 2i 0 0 0 0 0 0 1-i -1+i 0 1+i -1-i 0 0 complex lifted from C4≀C2 ρ18 2 -2 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 -√-2 0 0 -√-2 0 0 √-2 √-2 complex lifted from SD16 ρ19 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 4 -4 -4 0 0 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ21 4 4 -4 -4 0 0 0 0 0 0 2 0 0 0 -2 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 2i 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 -2i 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4

Smallest permutation representation of (C2×C4).D8
On 32 points
Generators in S32
(2 19)(4 21)(6 23)(8 17)(9 31)(11 25)(13 27)(15 29)
(1 26 18 12)(2 13 19 27)(3 14 20 28)(4 29 21 15)(5 30 22 16)(6 9 23 31)(7 10 24 32)(8 25 17 11)
(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 8 26 25 18 17 12 11)(2 32 13 7 19 10 27 24)(3 23 14 31 20 6 28 9)(4 30 29 22 21 16 15 5)

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

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

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

Matrix representation of (C2×C4).D8 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 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 4 13
,
 0 5 0 0 0 0 6 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 13 8 0 0 0 16 0 0 0 0 8 8 0 0
,
 1 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 15 0 0 4 0 0 0 0 0 2 2 0 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,4,0,0,0,0,0,13],[0,6,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,16,8,0,0,13,13,0,0,0,0,0,8,0,0],[1,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,2,0,0,1,1,0,0,0,0,0,15,0,0] >;

(C2×C4).D8 in GAP, Magma, Sage, TeX

(C_2\times C_4).D_8
% in TeX

G:=Group("(C2xC4).D8");
// GroupNames label

G:=SmallGroup(128,78);
// by ID

G=gap.SmallGroup(128,78);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,184,794,521,248,2804]);
// Polycyclic

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

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