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G = C8.5D8order 128 = 27

5th non-split extension by C8 of D8 acting via D8/C4=C22

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

Aliases: C8.5D8, D8.7D4, Q16.7D4, C42.144D4, M5(2).5C22, C8○D8.4C2, C4.69(C2×D8), C8.78(C2×D4), C8.C83C2, C8.3(C4○D4), (C2×C8).132D4, C4⋊Q1616C2, C8.17D44C2, Q32⋊C2.2C2, C4.57(C4⋊D4), C2.26(C4⋊D8), (C2×C8).237C23, (C4×C8).164C22, C4○D8.20C22, (C2×Q16).47C22, C22.26(C8⋊C22), C8.C4.20C22, (C2×C4).282(C2×D4), SmallGroup(128,946)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.5D8
C1C2C4C8C2×C8C4○D8C8○D8 — C8.5D8
C1C2C4C2×C8 — C8.5D8
C1C2C2×C4C4×C8 — C8.5D8
C1C2C2C2C2C4C4C2×C8 — C8.5D8

Generators and relations for C8.5D8
 G = < a,b,c | a8=1, b8=c2=a4, bab-1=a3, cac-1=a-1, cbc-1=b7 >

Subgroups: 172 in 75 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×5], C22, C22, C8 [×4], C8, C2×C4, C2×C4 [×4], D4 [×2], Q8 [×5], C16 [×2], C42, C4⋊C4 [×2], C2×C8 [×2], C2×C8, M4(2) [×2], D8, SD16, Q16, Q16 [×6], C2×Q8 [×2], C4○D4, C4×C8, C4≀C2, C8.C4, M5(2) [×2], SD32 [×2], Q32 [×2], C4⋊Q8, C8○D4, C2×Q16 [×2], C2×Q16, C4○D8, C8.17D4 [×2], C8.C8, C8○D8, C4⋊Q16, Q32⋊C2 [×2], C8.5D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C8.5D8

Character table of C8.5D8

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H16A16B16C16D
 size 1128224481616222244888888
ρ111111111111111111111111    trivial
ρ2111-111-1-1-11-111-1-1-1111-111-1    linear of order 2
ρ3111111111-1-111111111-1-1-1-1    linear of order 2
ρ4111-111-1-1-1-1111-1-1-11111-1-11    linear of order 2
ρ5111-11111-1-1-1111111-1-11111    linear of order 2
ρ6111111-1-11-1111-1-1-11-1-1-111-1    linear of order 2
ρ7111-11111-111111111-1-1-1-1-1-1    linear of order 2
ρ8111111-1-111-111-1-1-11-1-11-1-11    linear of order 2
ρ9222022-2-2000-2-2222-2000000    orthogonal lifted from D4
ρ1022-22-2200-20022000-2000000    orthogonal lifted from D4
ρ1122-2-2-220020022000-2000000    orthogonal lifted from D4
ρ1222202222000-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ1322-202-2000000022-2000-22-22    orthogonal lifted from D8
ρ1422-202-2000000022-20002-22-2    orthogonal lifted from D8
ρ1522-202-20000000-2-22000-2-222    orthogonal lifted from D8
ρ1622-202-20000000-2-2200022-2-2    orthogonal lifted from D8
ρ1722-20-2200000-2-20002-2i2i0000    complex lifted from C4○D4
ρ1822-20-2200000-2-200022i-2i0000    complex lifted from C4○D4
ρ194440-4-400000000000000000    orthogonal lifted from C8⋊C22
ρ204-40000-22000-222222-2200000000    symplectic faithful, Schur index 2
ρ214-400002-200022-2222-2200000000    symplectic faithful, Schur index 2
ρ224-40000-2200022-22-222200000000    symplectic faithful, Schur index 2
ρ234-400002-2000-2222-222200000000    symplectic faithful, Schur index 2

Smallest permutation representation of C8.5D8
On 32 points
Generators in S32
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)(17 23 29 19 25 31 21 27)(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 27 9 19)(2 18 10 26)(3 25 11 17)(4 32 12 24)(5 23 13 31)(6 30 14 22)(7 21 15 29)(8 28 16 20)

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

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

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

Matrix representation of C8.5D8 in GL4(𝔽7) generated by

5002
1334
1001
5366
,
3066
4523
1615
3155
,
3061
1102
1615
5642
G:=sub<GL(4,GF(7))| [5,1,1,5,0,3,0,3,0,3,0,6,2,4,1,6],[3,4,1,3,0,5,6,1,6,2,1,5,6,3,5,5],[3,1,1,5,0,1,6,6,6,0,1,4,1,2,5,2] >;

C8.5D8 in GAP, Magma, Sage, TeX

C_8._5D_8
% in TeX

G:=Group("C8.5D8");
// GroupNames label

G:=SmallGroup(128,946);
// by ID

G=gap.SmallGroup(128,946);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,512,422,2019,248,1684,438,242,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=b^7>;
// generators/relations

Export

Character table of C8.5D8 in TeX

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