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G = C8.1Q16  order 128 = 27

1st non-split extension by C8 of Q16 acting via Q16/C4=C22

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

Aliases: C8.1Q16, C8.28SD16, (C2×C8).80D4, C8.C8.2C2, C42.47(C2×C4), C42.C2.3C4, C8.5Q8.5C2, (C4×C8).135C22, C4.4(Q8⋊C4), C2.3(C4.6Q16), C22.15(C4.D4), (C2×C4).61(C22⋊C4), SmallGroup(128,98)

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

Derived series
C1 — C42 — C8.1Q16
Chief series
C1 — C2 — C22 — C2×C4 — C2×C8 — C4×C8 — C8.5Q8 — C8.1Q16
Lower central
C1 — C2 — C2×C4 — C42 — C8.1Q16
Upper central
C1 — C2 — C2×C4 — C4×C8 — C8.1Q16
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.1Q16

Generators and relations for C8.1Q16
 G = < a,b,c | a8=1, b8=a4, c2=a2b4, bab-1=a3, cac-1=a-1, cbc-1=a-1b7 >

C1
C2
2C2
C4
C4
C22
4C4
8C4
8C4
C8
C2×C4
C8
C8
C8
2C2×C4
4C2×C4
4C2×C4
C42
C2×C8
C2×C8
2C4⋊C4
2C4⋊C4
4C4⋊C4
4C16
4C4⋊C4
4C16
C4×C8
C42.C2
C42.C2
2M5(2)
2C2.D8
2C4.Q8
2M5(2)
C8.C8
C8.C8
C8.5Q8
C8.1Q16

Character table of C8.1Q16

 class 12A2B4A4B4C4D4E4F8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1122244161622224488888888
ρ111111111111111111111111    trivial
ρ21111111-1-1111111-11-111-1-11    linear of order 2
ρ31111111-1-11111111-11-1-111-1    linear of order 2
ρ4111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-1-1-1-1-i-iiiii-i-i    linear of order 4
ρ61111111-11-1-1-1-1-1-1i-i-iii-ii-i    linear of order 4
ρ711111111-1-1-1-1-1-1-1ii-i-i-i-iii    linear of order 4
ρ81111111-11-1-1-1-1-1-1-iii-i-ii-ii    linear of order 4
ρ922222-2-200-22-22-2200000000    orthogonal lifted from D4
ρ1022222-2-2002-22-22-200000000    orthogonal lifted from D4
ρ1122-22-20000-20-2020√20-√200√2-√20    symplectic lifted from Q16, Schur index 2
ρ1222-2-2200000-20-2020√20-√2√200-√2    symplectic lifted from Q16, Schur index 2
ρ1322-2-2200000-20-2020-√20√2-√200√2    symplectic lifted from Q16, Schur index 2
ρ1422-22-20000-20-2020-√20√200-√2√20    symplectic lifted from Q16, Schur index 2
ρ1522-22-200002020-20√-20√-200-√-2-√-20    complex lifted from SD16
ρ1622-2-22000002020-20√-20√-2-√-200-√-2    complex lifted from SD16
ρ1722-22-200002020-20-√-20-√-200√-2√-20    complex lifted from SD16
ρ1822-2-22000002020-20-√-20-√-2√-200√-2    complex lifted from SD16
ρ19444-4-4000000000000000000    orthogonal lifted from C4.D4
ρ204-4000-2i2i002√22√-2-2√2-2√-20000000000    complex faithful
ρ214-4000-2i2i00-2√2-2√-22√22√-20000000000    complex faithful
ρ224-40002i-2i00-2√22√-22√2-2√-20000000000    complex faithful
ρ234-40002i-2i002√2-2√-2-2√22√-20000000000    complex faithful

Smallest permutation representation of C8.1Q16
►On 32 points
Generators in S32
(1 22 13 18 9 30 5 26)(2 19 6 23 10 27 14 31)(3 24 15 20 11 32 7 28)(4 21 8 25 12 29 16 17)

(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)

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

Matrix representation of C8.1Q16 ►in GL4(𝔽17) generated by

31400
3300
001414
00314
,
0010
0001
12500
121200
,
1000
01600
00130
0004
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,14,3,0,0,14,14],[0,0,12,12,0,0,5,12,1,0,0,0,0,1,0,0],[1,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4] >;
 

C8.1Q16 in GAP, Magma, Sage, TeX 

C_8._1Q_{16}
 
% in TeX
 
G:=Group("C8.1Q16");
 
// GroupNames label
 
G:=SmallGroup(128,98);
 
// by ID
 
G=gap.SmallGroup(128,98);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,232,422,387,520,794,192,2804,1411,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a^2*b^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^-1*b^7>;
 
// generators/relations
 

Export

Subgroup lattice of C8.1Q16 in TeX
Character table of C8.1Q16 in TeX

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