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G = C8.29D8  order 128 = 27

6th non-split extension by C8 of D8 acting via D8/D4=C2

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

Aliases: C8.29D8, C8.3SD16, C4⋊Q8.3C4, (C2×C8).79D4, C8.C8⋊9C2, C8⋊5D4.1C2, C4⋊1D4.3C4, C42.46(C2×C4), C4.3(D4⋊C4), (C4×C8).134C22, C2.5(C4.D8), C22.14(C4.D4), (C2×C4).60(C22⋊C4), SmallGroup(128,91)

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

Derived series
C1 — C42 — C8.29D8
Chief series
C1 — C2 — C22 — C2×C4 — C2×C8 — C4×C8 — C8⋊5D4 — C8.29D8
Lower central
C1 — C2 — C2×C4 — C42 — C8.29D8
Upper central
C1 — C2 — C2×C4 — C4×C8 — C8.29D8
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.29D8

Generators and relations for C8.29D8
 G = < a,b,c | a8=1, b8=a4, c2=a, bab-1=a3, ac=ca, cbc-1=a5b7 >

C1
C2
2C2
16C2
C4
C22
C4
2C4
2C4
8C22
8C22
8C4
8C22
C2×C4
C8
C8
C8
C8
2C2×C4
4C2×C4
4C23
4D4
4Q8
4D4
4Q8
8D4
8D4
C42
C2×C8
C2×C8
2C2×Q8
2C2×D4
4C4⋊C4
4C16
4C2×D4
4SD16
4SD16
4SD16
4SD16
4C16
C4⋊Q8
C4×C8
C4⋊1D4
2M5(2)
2M5(2)
2C2×SD16
2C2×SD16
C8.C8
C8.C8
C8⋊5D4
C8.29D8

Character table of C8.29D8

 class 12A2B2C4A4B4C4D4E8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1121622441622224488888888
ρ111111111111111111111111    trivial
ρ2111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111-11111-1111111-111-1-111-1    linear of order 2
ρ4111-11111-11111111-1-111-1-11    linear of order 2
ρ511111111-1-1-1-1-1-1-1i-iii-ii-i-i    linear of order 4
ρ611111111-1-1-1-1-1-1-1-ii-i-ii-iii    linear of order 4
ρ7111-111111-1-1-1-1-1-1-i-ii-iii-ii    linear of order 4
ρ8111-111111-1-1-1-1-1-1ii-ii-i-ii-i    linear of order 4
ρ9222022-2-20-222-2-2200000000    orthogonal lifted from D4
ρ10222022-2-202-2-222-200000000    orthogonal lifted from D4
ρ1122-20-2200002200-2-√200√2-√200√2    orthogonal lifted from D8
ρ1222-20-2200002200-2√200-√2√200-√2    orthogonal lifted from D8
ρ1322-202-20002002-200-√2-√200√2√20    orthogonal lifted from D8
ρ1422-202-20002002-200√2√200-√2-√20    orthogonal lifted from D8
ρ1522-202-2000-200-2200√-2-√-200√-2-√-20    complex lifted from SD16
ρ1622-20-220000-2-2002√-200-√-2-√-200√-2    complex lifted from SD16
ρ1722-202-2000-200-2200-√-2√-200-√-2√-20    complex lifted from SD16
ρ1822-20-220000-2-2002-√-200√-2√-200-√-2    complex lifted from SD16
ρ194440-4-400000000000000000    orthogonal lifted from C4.D4
ρ204-40000-2202√-22√-2-2√-2-2√-20000000000    complex faithful
ρ214-400002-20-2√-22√-2-2√-22√-20000000000    complex faithful
ρ224-400002-202√-2-2√-22√-2-2√-20000000000    complex faithful
ρ234-40000-220-2√-2-2√-22√-22√-20000000000    complex faithful

Permutation representations of C8.29D8
►On 16 points - transitive group 16T346
Generators in S16
(1 11 5 15 9 3 13 7)(2 16 14 12 10 8 6 4)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(1 14 11 12 5 10 15 8 9 6 3 4 13 2 7 16)
 
G:=sub<Sym(16)| (1,11,5,15,9,3,13,7)(2,16,14,12,10,8,6,4), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,14,11,12,5,10,15,8,9,6,3,4,13,2,7,16)>;
 
G:=Group( (1,11,5,15,9,3,13,7)(2,16,14,12,10,8,6,4), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,14,11,12,5,10,15,8,9,6,3,4,13,2,7,16) );
 
G=PermutationGroup([[(1,11,5,15,9,3,13,7),(2,16,14,12,10,8,6,4)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,14,11,12,5,10,15,8,9,6,3,4,13,2,7,16)]])
 
G:=TransitiveGroup(16,346);
 

Matrix representation of C8.29D8 ►in GL4(𝔽3) generated by

1001
0020
0210
1000
,
0010
1000
0001
0120
,
0110
1002
0002
0120
G:=sub<GL(4,GF(3))| [1,0,0,1,0,0,2,0,0,2,1,0,1,0,0,0],[0,1,0,0,0,0,0,1,1,0,0,2,0,0,1,0],[0,1,0,0,1,0,0,1,1,0,0,2,0,2,2,0] >;
 

C8.29D8 in GAP, Magma, Sage, TeX 

C_8._{29}D_8
 
% in TeX
 
G:=Group("C8.29D8");
 
// GroupNames label
 
G:=SmallGroup(128,91);
 
// by ID
 
G=gap.SmallGroup(128,91);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,184,1690,416,2804,1411,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^5*b^7>;
 
// generators/relations
 

Export

Subgroup lattice of C8.29D8 in TeX
Character table of C8.29D8 in TeX

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