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## G = C2×C8.C8order 128 = 27

### Direct product of C2 and C8.C8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C8.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C4×C8 — C2×C8.C8
 Lower central C1 — C2 — C4 — C2×C8.C8
 Upper central C1 — C2×C8 — C22×C8 — C2×C8.C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C8.C8

Generators and relations for C2×C8.C8
G = < a,b,c | a2=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 108 in 84 conjugacy classes, 60 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×6], C23, C16 [×4], C42 [×2], C42, C2×C8 [×8], C2×C8 [×4], C22×C4, C22×C4, C4×C8 [×4], C2×C16 [×2], M5(2) [×4], M5(2) [×2], C2×C42, C22×C8 [×2], C8.C8 [×4], C2×C4×C8, C2×M5(2) [×2], C2×C8.C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C8.C8 [×2], C2×C4⋊C8, C2×C8.C8

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 8A ··· 8H 8I ··· 8T 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C4 C8 D4 Q8 M4(2) M4(2) C8.C8 kernel C2×C8.C8 C8.C8 C2×C4×C8 C2×M5(2) C4×C8 C2×C42 C22×C8 C2×C8 C2×C8 C2×C8 C2×C4 C23 C2 # reps 1 4 1 2 4 2 2 16 2 2 2 2 16

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

 16 0 0 0 16 0 0 0 16
,
 1 0 0 0 2 0 0 0 8
,
 13 0 0 0 0 1 0 9 0
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,2,0,0,0,8],[13,0,0,0,0,9,0,1,0] >;

C2×C8.C8 in GAP, Magma, Sage, TeX

C_2\times C_8.C_8
% in TeX

G:=Group("C2xC8.C8");
// GroupNames label

G:=SmallGroup(128,884);
// by ID

G=gap.SmallGroup(128,884);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,2019,248,102,124]);
// Polycyclic

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

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