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G = C42.14D4  order 128 = 27

14th non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial, rational

Aliases: C42.14D4, 2+ 1+4.5C22, (C2×Q8).35D4, C22⋊C4.2D4, C42⋊3C4⋊7C2, C2.25C2≀C22, (C2×D4).6C23, C23.18(C2×D4), C23.7D4.C2, D4.9D4.2C2, (C22×C4).105D4, C23.D4⋊4C2, C23⋊C4.5C22, C22.49C22≀C2, C4.D4.5C22, C4.4D4.20C22, C22.57C24⋊2C2, C22.D4.6C22, (C2×C4).18(C2×D4), SmallGroup(128,933)

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

Derived series
C1 — C2 — C2×D4 — C42.14D4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C22.D4 — C22.57C24 — C42.14D4
Lower central
C1 — C2 — C22 — C2×D4 — C42.14D4
Upper central
C1 — C2 — C22 — C2×D4 — C42.14D4
Jennings
C1 — C2 — C22 — C2×D4 — C42.14D4

Generators and relations for C42.14D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b, dad=a-1b-1, cbc-1=a2b-1, bd=db, dcd=b2c-1 >

Subgroups: 296 in 113 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C23⋊C4, C23⋊C4, C4.D4, C4≀C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C8.C22, 2+ 1+4, C23.D4, C42⋊3C4, D4.9D4, C23.7D4, C22.57C24, C42.14D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C2≀C22, C42.14D4

Character table of C42.14D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8
 size 11244848888888161616
ρ111111111111111111    trivial
ρ211111-11-11-1-1-1-11-111    linear of order 2
ρ311111-111-11-1-1-1-11-11    linear of order 2
ρ41111111-1-1-1111-1-1-11    linear of order 2
ρ511111111-11-1-11-1-11-1    linear of order 2
ρ611111-11-1-1-111-1-111-1    linear of order 2
ρ711111-1111111-11-1-1-1    linear of order 2
ρ81111111-11-1-1-1111-1-1    linear of order 2
ρ9222-2-202000-2200000    orthogonal lifted from D4
ρ102222-20-2020000-2000    orthogonal lifted from D4
ρ11222-2-2020002-200000    orthogonal lifted from D4
ρ12222-220-220-20000000    orthogonal lifted from D4
ρ132222-20-20-200002000    orthogonal lifted from D4
ρ14222-220-2-2020000000    orthogonal lifted from D4
ρ1544-4002000000-20000    orthogonal lifted from C2≀C22
ρ1644-400-200000020000    orthogonal lifted from C2≀C22
ρ178-8000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C42.14D4
►On 32 points
Generators in S32
(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 9 31 26)(2 10 32 27)(3 11 29 28)(4 12 30 25)(5 16 17 24)(6 13 18 21)(7 14 19 22)(8 15 20 23)

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

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

Matrix representation of C42.14D4 ►in GL8(𝔽17)

116111161166
611111111666
661161111116
666111111611
6116661166
1166611666
11111161111611
11116111111116
,
00100000
00010000
160000000
016000000
00000001
00000010
000001600
000016000
,
11611111161111
1166611666
11111161111611
6611666611
116111161166
116666111111
661161111116
111111666116
,
10000000
016000000
00100000
000160000
00001000
000001600
000000160
00000001

G:=sub<GL(8,GF(17))| [11,6,6,6,6,11,11,11,6,11,6,6,11,6,11,11,11,11,11,6,6,6,11,6,11,11,6,11,6,6,6,11,6,11,11,11,6,11,11,11,11,6,11,11,11,6,11,11,6,6,11,6,6,6,6,11,6,6,6,11,6,6,11,6],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[11,11,11,6,11,11,6,11,6,6,11,6,6,6,6,11,11,6,11,11,11,6,11,11,11,6,6,6,11,6,6,6,11,11,11,6,6,6,11,6,6,6,11,6,11,11,11,6,11,6,6,6,6,11,11,11,11,6,11,11,6,11,6,6],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;
 

C42.14D4 in GAP, Magma, Sage, TeX 

C_4^2._{14}D_4
 
% in TeX
 
G:=Group("C4^2.14D4");
 
// GroupNames label
 
G:=SmallGroup(128,933);
 
// by ID
 
G=gap.SmallGroup(128,933);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,723,352,297,1971,375,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b,d*a*d=a^-1*b^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d=b^2*c^-1>;
 
// generators/relations
 

Export

Character table of C42.14D4 in TeX

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