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## G = C2×C8○D8order 128 = 27

### Direct product of C2 and C8○D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C8○D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8○D4 — C2×C8○D8
 Lower central C1 — C2 — C4 — C2×C8○D8
 Upper central C1 — C2×C8 — C22×C8 — C2×C8○D8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8○D8

Generators and relations for C2×C8○D8
G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >

Subgroups: 348 in 236 conjugacy classes, 140 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×8], C22 [×3], C22 [×10], C8 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×2], C42, C2×C8 [×4], C2×C8 [×8], C2×C8 [×10], M4(2) [×4], M4(2) [×10], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C4×C8 [×4], C4≀C2 [×8], C8.C4 [×4], C2×C42, C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×8], C8○D4 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C2×C4×C8, C2×C4≀C2 [×2], C2×C8.C4, C8○D8 [×8], C2×C8○D4 [×2], C2×C4○D8, C2×C8○D8
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, C8○D8 [×2], C2×C4×D4, C2×C8○D8

Smallest permutation representation of C2×C8○D8
On 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(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 20 3 22 5 24 7 18)(2 21 4 23 6 17 8 19)(9 31 15 29 13 27 11 25)(10 32 16 30 14 28 12 26)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A ··· 8H 8I ··· 8T 8U ··· 8AB order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 C4○D4 C8○D8 kernel C2×C8○D8 C2×C4×C8 C2×C4≀C2 C2×C8.C4 C8○D8 C2×C8○D4 C2×C4○D8 C2×D8 C2×SD16 C2×Q16 C4○D8 C2×C8 C2×C4 C23 C2 # reps 1 1 2 1 8 2 1 2 4 2 8 4 2 2 16

Matrix representation of C2×C8○D8 in GL3(𝔽17) generated by

 16 0 0 0 16 0 0 0 16
,
 4 0 0 0 9 0 0 0 9
,
 1 0 0 0 9 0 0 0 2
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[4,0,0,0,9,0,0,0,9],[1,0,0,0,9,0,0,0,2],[1,0,0,0,0,1,0,1,0] >;

C2×C8○D8 in GAP, Magma, Sage, TeX

C_2\times C_8\circ D_8
% in TeX

G:=Group("C2xC8oD8");
// GroupNames label

G:=SmallGroup(128,1685);
// by ID

G=gap.SmallGroup(128,1685);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2804,1411,172,124]);
// Polycyclic

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

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