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G = D4.5D8  order 128 = 27

5th non-split extension by D4 of D8 acting via D8/C8=C2

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

Aliases: D4.5D8, Q8.5D8, C16.24D4, M4(2).36D4, M5(2).28C22, D4oC16:2C2, C4.43(C2xD8), (C2xSD32):2C2, C4oD4.28D4, C8.6(C4oD4), C8.107(C2xD4), D4.4D4.C2, C8.4Q8:7C2, M5(2):C2:8C2, D4.5D4:3C2, C8.17D4:8C2, C8oD4.9C22, C4.98(C4:D4), C2.26(C8:7D4), (C2xC8).240C23, (C2xC16).28C22, (C2xD8).50C22, C22.8(C4oD8), C8.C4.7C22, (C2xQ16).49C22, (C2xC4).45(C2xD4), SmallGroup(128,955)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2xC8 — D4.5D8
Chief series
C1 — C2 — C4 — C2xC4 — C2xC8 — C8oD4 — D4oC16 — D4.5D8
Lower central
C1 — C2 — C4 — C2xC8 — D4.5D8
Upper central
C1 — C2 — C2xC4 — C8oD4 — D4.5D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2xC8 — D4.5D8

Generators and relations for D4.5D8
 G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c7 >

Subgroups: 180 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C16, C16, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, C4.D4, C4.10D4, C8.C4, C2xC16, C2xC16, M5(2), M5(2), SD32, C8oD4, C2xD8, C2xQ16, C8:C22, C8.C22, M5(2):C2, C8.17D4, C8.4Q8, D4.4D4, D4.5D4, D4oC16, C2xSD32, D4.5D8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C4oD4, C4:D4, C2xD8, C4oD8, C8:7D4, D4.5D8

Character table of D4.5D8

 class 12A2B2C2D4A4B4C4D8A8B8C8D8E8F8G16A16B16C16D16E16F16G16H16I16J
 size 112416224162244416162222444444
ρ111111111111111111111111111    trivial
ρ2111-1-111-1-1111-1-11111111-1-1-1-11    linear of order 2
ρ3111-1-111-11111-1-1-11-1-1-1-1-11111-1    linear of order 2
ρ411111111-111111-11-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11111111111-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ6111-1111-1-1111-1-11-1-1-1-1-1-11111-1    linear of order 2
ρ7111-1111-11111-1-1-1-111111-1-1-1-11    linear of order 2
ρ81111-1111-111111-1-11111111111    linear of order 2
ρ922-2002-20022-20000-2-2-2-2200002    orthogonal lifted from D4
ρ10222202220-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ1122-2002-20022-200002222-20000-2    orthogonal lifted from D4
ρ12222-2022-20-2-2-222000000000000    orthogonal lifted from D4
ρ1322-220-22-200000000-√2√2-√2√2-√2-√2√2-√2√2√2    orthogonal lifted from D8
ρ1422-2-20-22200000000√2-√2√2-√2√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ1522-220-22-200000000√2-√2√2-√2√2√2-√2√2-√2-√2    orthogonal lifted from D8
ρ1622-2-20-22200000000-√2√2-√2√2-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1722200-2-2000002i-2i00√2-√2√2-√2-√2-√-2√-2√-2-√-2√2    complex lifted from C4oD8
ρ1822-2002-200-2-220000000002i2i-2i-2i0    complex lifted from C4oD4
ρ1922200-2-200000-2i2i00√2-√2√2-√2-√2√-2-√-2-√-2√-2√2    complex lifted from C4oD8
ρ2022200-2-2000002i-2i00-√2√2-√2√2√2√-2-√-2-√-2√-2-√2    complex lifted from C4oD8
ρ2122-2002-200-2-22000000000-2i-2i2i2i0    complex lifted from C4oD4
ρ2222200-2-200000-2i2i00-√2√2-√2√2√2-√-2√-2√-2-√-2-√2    complex lifted from C4oD8
ρ234-400000002√2-2√2000002ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ16000000    complex faithful
ρ244-400000002√2-2√2000002ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ169000000    complex faithful
ρ254-40000000-2√22√2000002ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ1611000000    complex faithful
ρ264-40000000-2√22√2000002ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ163000000    complex faithful

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

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

Matrix representation of D4.5D8 ►in GL4(F7) generated by

0651
3056
3361
1631
,
4320
6046
1145
0006
,
2045
5235
5214
4453
,
3633
6525
6102
2406
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[4,6,1,0,3,0,1,0,2,4,4,0,0,6,5,6],[2,5,5,4,0,2,2,4,4,3,1,5,5,5,4,3],[3,6,6,2,6,5,1,4,3,2,0,0,3,5,2,6] >;
 

D4.5D8 in GAP, Magma, Sage, TeX 

D_4._5D_8
 
% in TeX
 
G:=Group("D4.5D8");
 
// GroupNames label
 
G:=SmallGroup(128,955);
 
// by ID
 
G=gap.SmallGroup(128,955);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,1123,360,2804,718,172,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^7>;
 
// generators/relations
 

Export

Character table of D4.5D8 in TeX

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