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G = D4.3D8  order 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○C16⋊1C2, (C2×D16)⋊9C2, C4.41(C2×D8), C4○D4.26D4, C8.4(C4○D4), C8.105(C2×D4), C8.4Q8⋊5C2, D4.4D4⋊3C2, M5(2)⋊C2⋊7C2, C8○D4.7C22, C2.24(C8⋊7D4), 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

Derived series
C1 — C2×C8 — D4.3D8
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○C16 — D4.3D8
Lower central
C1 — C2 — C4 — C2×C8 — D4.3D8
Upper central
C1 — C2 — C2×C4 — C8○D4 — D4.3D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×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, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C16, C16, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C4.D4, C8.C4, C2×C16, C2×C16, M5(2), M5(2), D16, C8○D4, C2×D8, C8⋊C22, M5(2)⋊C2, C8.4Q8, D4.4D4, D4○C16, C2×D16, D4.3D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C4○D8, C8⋊7D4, 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
ρ1322-22002-2-20000000√2-√2√2-√2√2√2-√2√2-√2-√2    orthogonal lifted from D8
ρ1422-2-2002-220000000√2-√2√2-√2√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ1522-2-2002-220000000-√2√2-√2√2-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1622-22002-2-20000000-√2√2-√2√2-√2-√2√2-√2√2√2    orthogonal lifted from D8
ρ17222000-2-200002i-2i00-√2√2-√2√2√2√-2-√-2-√-2√-2-√2    complex lifted from C4○D8
ρ1822-2000-220-2-22000000000-2i-2i2i2i0    complex lifted from C4○D4
ρ19222000-2-20000-2i2i00√2-√2√2-√2-√2√-2-√-2-√-2√-2√2    complex lifted from C4○D8
ρ20222000-2-20000-2i2i00-√2√2-√2√2√2-√-2√-2√-2-√-2-√2    complex lifted from C4○D8
ρ21222000-2-200002i-2i00√2-√2√2-√2-√2-√-2√-2√-2-√-2√2    complex lifted from C4○D8
ρ2222-2000-220-2-220000000002i2i-2i-2i0    complex lifted from C4○D4
ρ234-400000002√2-2√200000-2ζ167+2ζ16-2ζ165+2ζ1632ζ167-2ζ162ζ165-2ζ163000000    orthogonal faithful
ρ244-40000000-2√22√2000002ζ165-2ζ163-2ζ167+2ζ16-2ζ165+2ζ1632ζ167-2ζ16000000    orthogonal faithful
ρ254-400000002√2-2√2000002ζ167-2ζ162ζ165-2ζ163-2ζ167+2ζ16-2ζ165+2ζ163000000    orthogonal faithful
ρ264-40000000-2√22√200000-2ζ165+2ζ1632ζ167-2ζ162ζ165-2ζ163-2ζ167+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 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(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)

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

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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