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G = C2×C8.Q8  order 128 = 27

Direct product of C2 and C8.Q8

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

Aliases: C2×C8.Q8, M5(2)⋊2C4, C23.38SD16, M5(2).15C22, (C2×C16)⋊1C4, C16⋊2(C2×C4), C8.30(C4⋊C4), (C2×C8).18Q8, C8.11(C2×Q8), (C2×C8).126D4, C4.6(C4.Q8), C8.58(C22×C4), (C2×C4).55SD16, C4.65(C2×SD16), (C2×C8).229C23, (C22×C4).339D4, (C2×M5(2)).4C2, C22.19(C2×SD16), C4.Q8.121C22, C22.16(C4.Q8), C8.C4.12C22, (C22×C8).238C22, C4.48(C2×C4⋊C4), (C2×C8).89(C2×C4), (C2×C4.Q8).9C2, C2.11(C2×C4.Q8), (C2×C4).274(C2×D4), (C2×C4).141(C4⋊C4), (C2×C8.C4).22C2, SmallGroup(128,886)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C8 — C2×C8.Q8
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×M5(2) — C2×C8.Q8
Lower central
C1 — C2 — C4 — C8 — C2×C8.Q8
Upper central
C1 — C22 — C22×C4 — C22×C8 — C2×C8.Q8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C8.Q8

Generators and relations for C2×C8.Q8
 G = < a,b,c,d | a2=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >

Subgroups: 140 in 76 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C4.Q8, C4.Q8, C8.C4, C8.C4, C2×C16, M5(2), C2×C4⋊C4, C22×C8, C2×M4(2), C8.Q8, C2×C4.Q8, C2×C8.C4, C2×M5(2), C2×C8.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4.Q8, C2×C4⋊C4, C2×SD16, C8.Q8, C2×C4.Q8, C2×C8.Q8

Smallest permutation representation of C2×C8.Q8
►On 32 points
Generators in S32
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 17)(15 18)(16 19)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J16A···16H
order12222244444444888888888816···16
size1111222222888822224488884···4

32 irreducible representations

dim1111111222224
type++++++-+
imageC1C2C2C2C2C4C4D4Q8D4SD16SD16C8.Q8
kernelC2×C8.Q8C8.Q8C2×C4.Q8C2×C8.C4C2×M5(2)C2×C16M5(2)C2×C8C2×C8C22×C4C2×C4C23C2
# reps1411144121624

Matrix representation of C2×C8.Q8 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0051200
005500
0000125
00001212
,
7150000
8100000
000010
000001
0051200
005500
,
1300000
640000
001000
0001600
0000125
000055

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[7,8,0,0,0,0,15,10,0,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,1,0,0,0,0,0,0,1,0,0],[13,6,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,12,5,0,0,0,0,5,5] >;
 

C2×C8.Q8 in GAP, Magma, Sage, TeX 

C_2\times C_8.Q_8
 
% in TeX
 
G:=Group("C2xC8.Q8");
 
// GroupNames label
 
G:=SmallGroup(128,886);
 
// by ID
 
G=gap.SmallGroup(128,886);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,1123,136,1411,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
 
// generators/relations
 

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