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G = D8.3D4order 128 = 27

3rd non-split extension by D8 of D4 acting via D4/C2=C22

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

Aliases: D8.3D4, D4.10D8, Q8.10D8, Q16.11D4, M4(2).18D4, M5(2)⋊4C22, D4○D8.C2, D4.C83C2, C8.68(C2×D4), C4.39(C2×D8), (C2×C16)⋊8C22, C4○D4.11D4, Q32⋊C22C2, C4.26C22≀C2, (C2×SD32)⋊12C2, M5(2)⋊C22C2, D8.C44C2, D4.5D41C2, C8○D4.3C22, C4○D8.7C22, (C2×C8).233C23, C8.C42C22, (C2×Q16)⋊11C22, (C2×D8).46C22, C2.34(C22⋊D8), C22.6(C8⋊C22), (C2×C4).41(C2×D4), SmallGroup(128,926)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8.3D4
C1C2C4C2×C4C2×C8C8○D4D4○D8 — D8.3D4
C1C2C4C2×C8 — D8.3D4
C1C2C2×C4C8○D4 — D8.3D4
C1C2C2C2C2C4C4C2×C8 — D8.3D4

Generators and relations for D8.3D4
 G = < a,b,c,d | a8=b2=1, c4=a6, d2=a4, bab=dad-1=a-1, cac-1=a5, cbc-1=a5b, dbd-1=ab, dcd-1=a6c3 >

Subgroups: 304 in 108 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×3], C22, C22 [×7], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4, D4 [×10], Q8, Q8 [×3], C23 [×3], C16 [×2], C2×C8, C2×C8, M4(2), M4(2) [×2], D8, D8 [×2], D8 [×3], SD16 [×4], Q16, Q16 [×3], C2×D4 [×6], C2×Q8, C4○D4, C4○D4 [×4], C4.10D4, C8.C4, C2×C16, M5(2), SD32 [×3], Q32, C8○D4, C2×D8, C2×D8, C2×Q16, C4○D8, C4○D8, C8⋊C22 [×3], C8.C22, 2+ 1+4, D4.C8, D8.C4, M5(2)⋊C2, D4.5D4, C2×SD32, Q32⋊C2, D4○D8, D8.3D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8.3D4

Character table of D8.3D4

 class 12A2B2C2D2E2F4A4B4C4D4E8A8B8C8D8E16A16B16C16D16E16F
 size 1124888224816224816444488
ρ111111111111111111111111    trivial
ρ2111-11-1-111-11-1111-111111-1-1    linear of order 2
ρ3111-11-1-111-111111-1-1-1-1-1-111    linear of order 2
ρ411111111111-11111-1-1-1-1-1-1-1    linear of order 2
ρ5111-1-11111-1-1-1111-11-1-1-1-111    linear of order 2
ρ61111-1-1-1111-1111111-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-1111-1-11111-1111111    linear of order 2
ρ8111-1-11111-1-11111-1-11111-1-1    linear of order 2
ρ922-200-22-22000-2-2200000000    orthogonal lifted from D4
ρ10222-200022-200-2-2-220000000    orthogonal lifted from D4
ρ1122-20200-220-2022-200000000    orthogonal lifted from D4
ρ1222-2002-2-22000-2-2200000000    orthogonal lifted from D4
ρ1322-20-200-2202022-200000000    orthogonal lifted from D4
ρ14222200022200-2-2-2-20000000    orthogonal lifted from D4
ρ1522-2-20002-220000000-22-22-22    orthogonal lifted from D8
ρ1622-220002-2-20000000-22-222-2    orthogonal lifted from D8
ρ1722-2-20002-2200000002-22-22-2    orthogonal lifted from D8
ρ1822-220002-2-200000002-22-2-22    orthogonal lifted from D8
ρ194440000-4-400000000000000    orthogonal lifted from C8⋊C22
ρ204-4000000000022-22000ζ16716516316ζ1613161116716ζ161516131611169ζ161516916516300    complex faithful
ρ214-40000000000-2222000ζ1615169165163ζ16716516316ζ1613161116716ζ16151613161116900    complex faithful
ρ224-40000000000-2222000ζ1613161116716ζ161516131611169ζ1615169165163ζ1671651631600    complex faithful
ρ234-4000000000022-22000ζ161516131611169ζ1615169165163ζ16716516316ζ161316111671600    complex faithful

Smallest permutation representation of D8.3D4
On 32 points
Generators in S32
(1 17 13 29 9 25 5 21)(2 26 14 22 10 18 6 30)(3 19 15 31 11 27 7 23)(4 28 16 24 12 20 8 32)
(1 7)(2 20)(3 5)(4 18)(6 32)(8 30)(9 15)(10 28)(11 13)(12 26)(14 24)(16 22)(17 27)(19 25)(21 23)(29 31)
(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 2 9 10)(3 16 11 8)(4 7 12 15)(5 14 13 6)(17 30 25 22)(18 21 26 29)(19 28 27 20)(23 24 31 32)

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

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

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

Matrix representation of D8.3D4 in GL4(𝔽7) generated by

2224
1436
1062
5432
,
5446
6346
6602
2016
,
3261
3634
5401
2025
,
3532
1442
1636
5524
G:=sub<GL(4,GF(7))| [2,1,1,5,2,4,0,4,2,3,6,3,4,6,2,2],[5,6,6,2,4,3,6,0,4,4,0,1,6,6,2,6],[3,3,5,2,2,6,4,0,6,3,0,2,1,4,1,5],[3,1,1,5,5,4,6,5,3,4,3,2,2,2,6,4] >;

D8.3D4 in GAP, Magma, Sage, TeX

D_8._3D_4
% in TeX

G:=Group("D8.3D4");
// GroupNames label

G:=SmallGroup(128,926);
// by ID

G=gap.SmallGroup(128,926);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,1123,570,360,2804,718,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^6,d^2=a^4,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^5*b,d*b*d^-1=a*b,d*c*d^-1=a^6*c^3>;
// generators/relations

Export

Character table of D8.3D4 in TeX

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