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G = C8.24D8order 128 = 27

1st non-split extension by C8 of D8 acting via D8/D4=C2

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

Aliases: C8.24D8, C8.1SD16, (C2×C8).77D4, C8.C88C2, C41D4.2C4, C84D4.12C2, C42.44(C2×C4), C4.1(D4⋊C4), (C4×C8).132C22, C2.3(C4.D8), C22.12(C4.D4), (C2×C4).58(C22⋊C4), SmallGroup(128,89)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8.24D8
C1C2C22C2×C4C2×C8C4×C8C84D4 — C8.24D8
C1C2C2×C4C42 — C8.24D8
C1C2C2×C4C4×C8 — C8.24D8
C1C2C2C2C2C2×C4C2×C4C4×C8 — C8.24D8

Generators and relations for C8.24D8
 G = < a,b,c | a8=1, b8=a4, c2=a, bab-1=a-1, ac=ca, cbc-1=a5b7 >

2C2
16C2
16C2
2C4
2C4
8C22
8C22
8C22
8C22
8C22
8C22
2C2×C4
4D4
4D4
4C23
4D4
4D4
4C23
8D4
8D4
8D4
8D4
2C2×D4
2C2×D4
4D8
4D8
4C2×D4
4D8
4C16
4C2×D4
4D8
4C16
2M5(2)
2C2×D8
2C2×D8
2M5(2)

Character table of C8.24D8

 class 12A2B2C2D4A4B4C4D8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1121616224422224488888888
ρ111111111111111111111111    trivial
ρ2111-1-11111111111-11-11-11-11    linear of order 2
ρ3111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-1-111111111111-11-11-11-1    linear of order 2
ρ5111-111111-1-1-1-1-1-1i-i-ii-iii-i    linear of order 4
ρ61111-11111-1-1-1-1-1-1-i-iiiii-i-i    linear of order 4
ρ7111-111111-1-1-1-1-1-1-iii-ii-i-ii    linear of order 4
ρ81111-11111-1-1-1-1-1-1ii-i-i-i-iii    linear of order 4
ρ92220022-2-222-2-2-2200000000    orthogonal lifted from D4
ρ102220022-2-2-2-2222-200000000    orthogonal lifted from D4
ρ1122-200-22000022-200-2020-202    orthogonal lifted from D8
ρ1222-2002-20022000-2-20-202020    orthogonal lifted from D8
ρ1322-200-22000022-20020-2020-2    orthogonal lifted from D8
ρ1422-2002-20022000-22020-20-20    orthogonal lifted from D8
ρ1522-200-220000-2-2200--20--20-20-2    complex lifted from SD16
ρ1622-2002-200-2-20002--20-20--20-20    complex lifted from SD16
ρ1722-2002-200-2-20002-20--20-20--20    complex lifted from SD16
ρ1822-200-220000-2-2200-20-20--20--2    complex lifted from SD16
ρ1944400-4-40000000000000000    orthogonal lifted from C4.D4
ρ204-4000002-2-2222-22220000000000    orthogonal faithful
ρ214-400000-22-222222-220000000000    orthogonal faithful
ρ224-4000002-222-2222-220000000000    orthogonal faithful
ρ234-400000-2222-22-22220000000000    orthogonal faithful

Permutation representations of C8.24D8
On 16 points - transitive group 16T356
Generators in S16
(1 11 5 15 9 3 13 7)(2 8 14 4 10 16 6 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(1 4 11 10 5 16 15 6 9 12 3 2 13 8 7 14)

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

G:=Group( (1,11,5,15,9,3,13,7)(2,8,14,4,10,16,6,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,4,11,10,5,16,15,6,9,12,3,2,13,8,7,14) );

G=PermutationGroup([(1,11,5,15,9,3,13,7),(2,8,14,4,10,16,6,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,4,11,10,5,16,15,6,9,12,3,2,13,8,7,14)])

G:=TransitiveGroup(16,356);

Matrix representation of C8.24D8 in GL4(𝔽7) generated by

5026
6556
6112
2262
,
5305
0605
5131
3550
,
5305
0102
5660
3262
G:=sub<GL(4,GF(7))| [5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[5,0,5,3,3,6,1,5,0,0,3,5,5,5,1,0],[5,0,5,3,3,1,6,2,0,0,6,6,5,2,0,2] >;

C8.24D8 in GAP, Magma, Sage, TeX

C_8._{24}D_8
% in TeX

G:=Group("C8.24D8");
// GroupNames label

G:=SmallGroup(128,89);
// by ID

G=gap.SmallGroup(128,89);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,184,1690,192,2804,1411,172,4037]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^7>;
// generators/relations

Export

Subgroup lattice of C8.24D8 in TeX
Character table of C8.24D8 in TeX

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