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G = C8.3D8order 128 = 27

3rd non-split extension by C8 of D8 acting via D8/C4=C22

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

Aliases: C8.3D8, D8.6D4, Q16.6D4, C42.142D4, M5(2).3C22, C8○D83C2, C4.67(C2×D8), C8.76(C2×D4), C8.C81C2, C8.1(C4○D4), Q32⋊C24C2, (C2×C8).130D4, M5(2)⋊C23C2, C8.17D43C2, C16⋊C22.2C2, C8.12D412C2, C4.55(C4⋊D4), C2.24(C4⋊D8), (C2×C8).235C23, (C4×C8).162C22, C4○D8.18C22, (C2×D8).47C22, (C2×Q16).46C22, C22.24(C8⋊C22), C8.C4.18C22, (C2×C4).280(C2×D4), SmallGroup(128,944)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.3D8
C1C2C4C8C2×C8C4○D8C8○D8 — C8.3D8
C1C2C4C2×C8 — C8.3D8
C1C2C2×C4C4×C8 — C8.3D8
C1C2C2C2C2C4C4C2×C8 — C8.3D8

Generators and relations for C8.3D8
 G = < a,b,c | a8=c2=1, b8=a4, bab-1=a-1, cac=a3, cbc=a4b7 >

Subgroups: 204 in 77 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], C22, C22 [×4], C8 [×4], C8, C2×C4, C2×C4 [×3], D4 [×4], Q8 [×3], C23, C16 [×2], C42, C22⋊C4 [×2], C2×C8 [×2], C2×C8, M4(2) [×2], D8, D8 [×2], SD16 [×3], Q16, Q16 [×2], C2×D4, C2×Q8, C4○D4, C4×C8, C4≀C2, C8.C4, M5(2) [×2], D16, SD32 [×2], Q32, C4.4D4, C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, M5(2)⋊C2, C8.17D4, C8.C8, C8○D8, C8.12D4, C16⋊C22, Q32⋊C2, C8.3D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C8.3D8

Character table of C8.3D8

 class 12A2B2C2D4A4B4C4D4E4F8A8B8C8D8E8F8G8H16A16B16C16D
 size 1128162244816222244888888
ρ111111111111111111111111    trivial
ρ21111-111111-111111111-1-1-1-1    linear of order 2
ρ31111111-1-11-11-1-11-11-1-1-11-11    linear of order 2
ρ41111-111-1-1111-1-11-11-1-11-11-1    linear of order 2
ρ5111-1-11111-1-1111111-1-11111    linear of order 2
ρ6111-111111-11111111-1-1-1-1-1-1    linear of order 2
ρ7111-1-111-1-1-111-1-11-1111-11-11    linear of order 2
ρ8111-1111-1-1-1-11-1-11-11111-11-1    linear of order 2
ρ922-220-2200-2020020-2000000    orthogonal lifted from D4
ρ1022-2-20-22002020020-2000000    orthogonal lifted from D4
ρ112220022-2-200-222-22-2000000    orthogonal lifted from D4
ρ1222200222200-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ1322-2002-200000-2-202000-2-222    orthogonal lifted from D8
ρ1422-2002-200000220-2000-222-2    orthogonal lifted from D8
ρ1522-2002-200000220-20002-2-22    orthogonal lifted from D8
ρ1622-2002-200000-2-20200022-2-2    orthogonal lifted from D8
ρ1722-200-220000-200-2022i-2i0000    complex lifted from C4○D4
ρ1822-200-220000-200-202-2i2i0000    complex lifted from C4○D4
ρ1944400-4-40000000000000000    orthogonal lifted from C8⋊C22
ρ204-4000002i-2i00-222-2-2-22200000000    complex faithful
ρ214-4000002i-2i0022-2-22-2-2200000000    complex faithful
ρ224-400000-2i2i00-22-2-22-22200000000    complex faithful
ρ234-400000-2i2i00222-2-2-2-2200000000    complex faithful

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

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

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

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

Matrix representation of C8.3D8 in GL4(𝔽17) generated by

10700
5000
00710
00120
,
0006
0030
1000
11600
,
1000
11600
0006
0030
G:=sub<GL(4,GF(17))| [10,5,0,0,7,0,0,0,0,0,7,12,0,0,10,0],[0,0,1,1,0,0,0,16,0,3,0,0,6,0,0,0],[1,1,0,0,0,16,0,0,0,0,0,3,0,0,6,0] >;

C8.3D8 in GAP, Magma, Sage, TeX

C_8._3D_8
% in TeX

G:=Group("C8.3D8");
// GroupNames label

G:=SmallGroup(128,944);
// by ID

G=gap.SmallGroup(128,944);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,64,422,2019,248,1684,438,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of C8.3D8 in TeX

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