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G = C2×C8.C8  order 128 = 27

Direct product of C2 and C8.C8

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

Aliases: C2×C8.C8, C23.34M4(2), M5(2).22C22, (C4×C8).29C4, (C2×C8).14C8, C8.18(C2×C8), C8.46(C4⋊C4), C4.29(C4⋊C8), (C2×C8).63Q8, C8.26(C2×Q8), C4○(C8.C8), C8○(C8.C8), C8.136(C2×D4), (C2×C8).404D4, C22.7(C4⋊C8), (C22×C8).37C4, C4.28(C22×C8), (C2×C42).49C4, (C2×C8).605C23, (C4×C8).420C22, C42.329(C2×C4), (C2×C4).83M4(2), (C2×M5(2)).25C2, (C22×C8).580C22, C22.23(C2×M4(2)), (C2×C4×C8).50C2, C2.15(C2×C4⋊C8), C4.81(C2×C4⋊C4), (C2×C4).89(C2×C8), (C2×C8).252(C2×C4), (C2×C4).139(C4⋊C4), (C2×C4).560(C22×C4), (C22×C4).488(C2×C4), SmallGroup(128,884)

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, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, M5(2), M5(2), C2×C42, C22×C8, C8.C8, C2×C4×C8, C2×M5(2), C2×C8.C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C8.C8, C2×C4⋊C8, C2×C8.C8

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

(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)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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( (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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([[(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(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 2A2B2C2D2E4A4B4C4D4E···4N8A···8H8I···8T16A···16P
order12222244444···48···88···816···16
size11112211112···21···12···24···4

56 irreducible representations

dim1111111122222
type+++++-
imageC1C2C2C2C4C4C4C8D4Q8M4(2)M4(2)C8.C8
kernelC2×C8.C8C8.C8C2×C4×C8C2×M5(2)C4×C8C2×C42C22×C8C2×C8C2×C8C2×C8C2×C4C23C2
# reps141242216222216

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

1600
0160
0016
,
100
020
008
,
1300
001
090
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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