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G = D4.3D8order 128 = 27

3rd non-split extension by D4 of D8 acting via D8/C8=C2

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

Aliases: D4.3D8, Q8.3D8, C16.20D4, M4(2).34D4, M5(2).26C22, D4○C161C2, (C2×D16)⋊9C2, C4.41(C2×D8), C4○D4.26D4, C8.4(C4○D4), C8.105(C2×D4), C8.4Q85C2, D4.4D43C2, M5(2)⋊C27C2, C8○D4.7C22, C2.24(C87D4), C4.96(C4⋊D4), (C2×C16).26C22, (C2×C8).238C23, (C2×D8).49C22, C22.6(C4○D8), C8.C4.5C22, (C2×C4).43(C2×D4), SmallGroup(128,953)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D4.3D8
C1C2C4C2×C4C2×C8C8○D4D4○C16 — D4.3D8
C1C2C4C2×C8 — D4.3D8
C1C2C2×C4C8○D4 — D4.3D8
C1C2C2C2C2C4C4C2×C8 — D4.3D8

Generators and relations for D4.3D8
 G = < a,b,c,d | a4=b2=d2=1, c8=a2, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c7 >

Subgroups: 212 in 74 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C8 [×2], C8 [×3], C2×C4, C2×C4, D4, D4 [×5], Q8, C23 [×2], C16 [×2], C16, C2×C8, C2×C8, M4(2), M4(2) [×3], D8 [×6], SD16 [×2], C2×D4 [×2], C4○D4, C4.D4 [×2], C8.C4 [×2], C2×C16, C2×C16, M5(2), M5(2), D16 [×2], C8○D4, C2×D8 [×2], C8⋊C22 [×2], M5(2)⋊C2 [×2], C8.4Q8, D4.4D4 [×2], D4○C16, C2×D16, D4.3D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4, D4.3D8

Character table of D4.3D8

 class 12A2B2C2D2E4A4B4C8A8B8C8D8E8F8G16A16B16C16D16E16F16G16H16I16J
 size 112416162242244416162222444444
ρ111111111111111111111111111    trivial
ρ2111-1-1-111-1111-1-11111111-1-1-1-11    linear of order 2
ρ31111-1-111111111-1-11111111111    linear of order 2
ρ4111-11111-1111-1-1-1-111111-1-1-1-11    linear of order 2
ρ51111-11111111111-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ6111-11-111-1111-1-11-1-1-1-1-1-11111-1    linear of order 2
ρ711111-111111111-11-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ8111-1-1111-1111-1-1-11-1-1-1-1-11111-1    linear of order 2
ρ922-2000-22022-20000-2-2-2-2200002    orthogonal lifted from D4
ρ1022-2000-22022-200002222-20000-2    orthogonal lifted from D4
ρ11222200222-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ12222-20022-2-2-2-222000000000000    orthogonal lifted from D4
ρ1322-22002-2-200000002-22-222-22-2-2    orthogonal lifted from D8
ρ1422-2-2002-2200000002-22-22-22-22-2    orthogonal lifted from D8
ρ1522-2-2002-220000000-22-22-22-22-22    orthogonal lifted from D8
ρ1622-22002-2-20000000-22-22-2-22-222    orthogonal lifted from D8
ρ17222000-2-200002i-2i00-22-222-2--2--2-2-2    complex lifted from C4○D8
ρ1822-2000-220-2-22000000000-2i-2i2i2i0    complex lifted from C4○D4
ρ19222000-2-20000-2i2i002-22-2-2-2--2--2-22    complex lifted from C4○D8
ρ20222000-2-20000-2i2i00-22-222--2-2-2--2-2    complex lifted from C4○D8
ρ21222000-2-200002i-2i002-22-2-2--2-2-2--22    complex lifted from C4○D8
ρ2222-2000-220-2-220000000002i2i-2i-2i0    complex lifted from C4○D4
ρ234-4000000022-2200000-2ζ167+2ζ16-2ζ165+2ζ163167-2ζ16165-2ζ163000000    orthogonal faithful
ρ244-40000000-222200000165-2ζ163-2ζ167+2ζ16-2ζ165+2ζ163167-2ζ16000000    orthogonal faithful
ρ254-4000000022-2200000167-2ζ16165-2ζ163-2ζ167+2ζ16-2ζ165+2ζ163000000    orthogonal faithful
ρ264-40000000-222200000-2ζ165+2ζ163167-2ζ16165-2ζ163-2ζ167+2ζ16000000    orthogonal faithful

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

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

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

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

Matrix representation of D4.3D8 in GL4(𝔽17) generated by

01600
1000
161601
116160
,
145015
3520
146123
3553
,
4600
11400
71046
1010114
,
141400
14300
1216143
61233
G:=sub<GL(4,GF(17))| [0,1,16,1,16,0,16,16,0,0,0,16,0,0,1,0],[14,3,14,3,5,5,6,5,0,2,12,5,15,0,3,3],[4,11,7,10,6,4,10,10,0,0,4,11,0,0,6,4],[14,14,12,6,14,3,16,12,0,0,14,3,0,0,3,3] >;

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

D_4._3D_8
% in TeX

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

G:=SmallGroup(128,953);
// by ID

G=gap.SmallGroup(128,953);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,1123,360,2804,718,172,4037,124]);
// Polycyclic

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

Export

Character table of D4.3D8 in TeX

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