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G = C8.25D8  order 128 = 27

2nd non-split extension by C8 of D8 acting via D8/D4=C2

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

Aliases: C8.25D8, C8.2SD16, C4⋊Q8.2C4, (C2×C8).78D4, C8.C8.1C2, C42.45(C2×C4), C4.2(D4⋊C4), (C4×C8).133C22, C4⋊Q16.12C2, C2.4(C4.D8), C22.13(C4.D4), (C2×C4).59(C22⋊C4), SmallGroup(128,90)

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

Derived series
C1 — C42 — C8.25D8
Chief series
C1 — C2 — C22 — C2×C4 — C2×C8 — C4×C8 — C4⋊Q16 — C8.25D8
Lower central
C1 — C2 — C2×C4 — C42 — C8.25D8
Upper central
C1 — C2 — C2×C4 — C4×C8 — C8.25D8
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.25D8

Generators and relations for C8.25D8
 G = < a,b,c | a8=1, b8=a4, c2=a5, bab-1=a-1, ac=ca, cbc-1=a5b7 >

C1
C2
2C2
C4
C22
C4
2C4
2C4
8C4
8C4
C2×C4
C8
C8
C8
C8
2C2×C4
4C2×C4
4Q8
4Q8
4Q8
4Q8
4C2×C4
C42
C2×C8
C2×C8
2C2×Q8
2C2×Q8
4Q16
4Q16
4Q16
4C4⋊C4
4C4⋊C4
4C16
4C16
4Q16
C4×C8
C4⋊Q8
C4⋊Q8
2C2×Q16
2M5(2)
2C2×Q16
2M5(2)
C4⋊Q16
C8.C8
C8.C8
C8.25D8

Character table of C8.25D8

 class 12A2B4A4B4C4D4E4F8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1122244161622224488888888
ρ111111111111111111111111    trivial
ρ21111111-1-1111111111-1-1-11-1    linear of order 2
ρ31111111-1-1111111-1-1-1111-11    linear of order 2
ρ4111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111111-11-1-1-1-1-1-1i-i-iii-ii-i    linear of order 4
ρ611111111-1-1-1-1-1-1-1i-i-i-i-iiii    linear of order 4
ρ71111111-11-1-1-1-1-1-1-iii-i-ii-ii    linear of order 4
ρ811111111-1-1-1-1-1-1-1-iiiii-i-i-i    linear of order 4
ρ922222-2-200-22-22-2200000000    orthogonal lifted from D4
ρ1022222-2-2002-22-22-200000000    orthogonal lifted from D4
ρ1122-2-22000002020-2000-√2√2-√20√2    orthogonal lifted from D8
ρ1222-22-200002020-20√2√2-√2000-√20    orthogonal lifted from D8
ρ1322-22-200002020-20-√2-√2√2000√20    orthogonal lifted from D8
ρ1422-2-22000002020-2000√2-√2√20-√2    orthogonal lifted from D8
ρ1522-22-20000-20-2020√-2-√-2√-2000-√-20    complex lifted from SD16
ρ1622-2-2200000-20-202000-√-2√-2√-20-√-2    complex lifted from SD16
ρ1722-22-20000-20-2020-√-2√-2-√-2000√-20    complex lifted from SD16
ρ1822-2-2200000-20-202000√-2-√-2-√-20√-2    complex lifted from SD16
ρ19444-4-4000000000000000000    orthogonal lifted from C4.D4
ρ204-4000-2200-2√22√22√2-2√20000000000    symplectic faithful, Schur index 2
ρ214-40002-200-2√2-2√22√22√20000000000    symplectic faithful, Schur index 2
ρ224-40002-2002√22√2-2√2-2√20000000000    symplectic faithful, Schur index 2
ρ234-4000-22002√2-2√2-2√22√20000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C8.25D8
►On 32 points
Generators in S32
(1 3 5 7 9 11 13 15)(2 16 14 12 10 8 6 4)(17 19 21 23 25 27 29 31)(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)

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

Matrix representation of C8.25D8 ►in GL4(𝔽7) generated by

2532
6235
6602
2564
,
6620
4533
4201
2213
,
3315
3233
5445
2565
G:=sub<GL(4,GF(7))| [2,6,6,2,5,2,6,5,3,3,0,6,2,5,2,4],[6,4,4,2,6,5,2,2,2,3,0,1,0,3,1,3],[3,3,5,2,3,2,4,5,1,3,4,6,5,3,5,5] >;
 

C8.25D8 in GAP, Magma, Sage, TeX 

C_8._{25}D_8
 
% in TeX
 
G:=Group("C8.25D8");
 
// GroupNames label
 
G:=SmallGroup(128,90);
 
// by ID
 
G=gap.SmallGroup(128,90);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,184,1690,192,2804,1411,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a^5,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^7>;
 
// generators/relations
 

Export

Subgroup lattice of C8.25D8 in TeX
Character table of C8.25D8 in TeX

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