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G = C42⋊9D4  order 128 = 27

3rd semidirect product of C42 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42⋊9D4, M4(2)⋊1D4, C24.29D4, C4⋊C4⋊6D4, (C2×D4)⋊6D4, (C2×Q8)⋊6D4, C4.1C22≀C2, C42⋊6C4⋊6C2, C4.39(C4⋊D4), C4.29(C4⋊1D4), C23.577(C2×D4), C2.30(D4⋊4D4), C22.29C24⋊1C2, C2.7(C23⋊2D4), C22.195C22≀C2, C23.37D4⋊27C2, C22.55(C4⋊D4), (C22×C4).708C23, (C2×C42).341C22, (C22×D4).58C22, C42⋊C2.46C22, (C2×M4(2)).11C22, (C2×C4≀C2)⋊23C2, (C2×C4⋊1D4)⋊1C2, (C2×C8⋊C22)⋊1C2, (C2×C4.D4)⋊1C2, (C2×C4).72(C4○D4), (C2×C4).1023(C2×D4), (C2×C4○D4).43C22, SmallGroup(128,734)

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

Derived series
C1 — C22×C4 — C42⋊9D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C2×C4⋊1D4 — C42⋊9D4
Lower central
C1 — C2 — C22×C4 — C42⋊9D4
Upper central
C1 — C22 — C22×C4 — C42⋊9D4
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊9D4

Generators and relations for C42⋊9D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 656 in 239 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, C4.D4, D4⋊C4, C4≀C2, C2×C42, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C22×D4, C2×C4○D4, C42⋊6C4, C2×C4.D4, C23.37D4, C2×C4≀C2, C2×C4⋊1D4, C22.29C24, C2×C8⋊C22, C42⋊9D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C23⋊2D4, D4⋊4D4, C42⋊9D4

Character table of C42⋊9D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112288888222244448888888
ρ111111111111111111111111111    trivial
ρ21111111-11-111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ31111111-1-1-1-111111111-1-11-11-11    linear of order 2
ρ411111111-11-11111-1-1-1-1-1-111-11-1    linear of order 2
ρ5111111-1-11-111111-1-1-1-1-1-1-11111    linear of order 2
ρ6111111-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ7111111-11-11-11111-1-1-1-111-1-11-11    linear of order 2
ρ8111111-1-1-1-1-11111111111-11-11-1    linear of order 2
ρ92222-2-2-2000022-2-200000020000    orthogonal lifted from D4
ρ102222-2-200000-2-2220000-2200000    orthogonal lifted from D4
ρ112-2-222-20-2020-22-2200000000000    orthogonal lifted from D4
ρ122-2-22-2200000-222-2000000020-20    orthogonal lifted from D4
ρ132-2-222-2020-20-22-2200000000000    orthogonal lifted from D4
ρ142222-2-22000022-2-2000000-20000    orthogonal lifted from D4
ρ152222220020-2-2-2-2-200000000000    orthogonal lifted from D4
ρ162222-2-200000-2-22200002-200000    orthogonal lifted from D4
ρ172-2-22-2200000-222-20000000-2020    orthogonal lifted from D4
ρ182-2-222-2000002-22-22-22-20000000    orthogonal lifted from D4
ρ1922222200-202-2-2-2-200000000000    orthogonal lifted from D4
ρ202-2-222-2000002-22-2-22-220000000    orthogonal lifted from D4
ρ212-2-22-22000002-2-22000000002i0-2i    complex lifted from C4○D4
ρ222-2-22-22000002-2-2200000000-2i02i    complex lifted from C4○D4
ρ234-44-400000000000-222-20000000    orthogonal lifted from D4⋊4D4
ρ2444-4-400000000000-2-2220000000    orthogonal lifted from D4⋊4D4
ρ2544-4-40000000000022-2-20000000    orthogonal lifted from D4⋊4D4
ρ264-44-4000000000002-2-220000000    orthogonal lifted from D4⋊4D4

Permutation representations of C42⋊9D4
►On 16 points - transitive group 16T408
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

(1 8 2 7)(3 6 4 5)(9 12 11 10)(13 16 15 14)

(1 12 3 14)(2 10 4 16)(5 15 7 9)(6 13 8 11)

(1 9)(2 11)(3 15)(4 13)(5 14)(6 16)(7 12)(8 10)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,2,7)(3,6,4,5)(9,12,11,10)(13,16,15,14), (1,12,3,14)(2,10,4,16)(5,15,7,9)(6,13,8,11), (1,9)(2,11)(3,15)(4,13)(5,14)(6,16)(7,12)(8,10)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,2,7)(3,6,4,5)(9,12,11,10)(13,16,15,14), (1,12,3,14)(2,10,4,16)(5,15,7,9)(6,13,8,11), (1,9)(2,11)(3,15)(4,13)(5,14)(6,16)(7,12)(8,10) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,8,2,7),(3,6,4,5),(9,12,11,10),(13,16,15,14)], [(1,12,3,14),(2,10,4,16),(5,15,7,9),(6,13,8,11)], [(1,9),(2,11),(3,15),(4,13),(5,14),(6,16),(7,12),(8,10)]])
 
G:=TransitiveGroup(16,408);
 

Matrix representation of C42⋊9D4 ►in GL6(ℤ)

0-10000
100000
001000
000100
000012
0000-1-1
,
-100000
0-10000
00-1-200
001100
000012
0000-1-1
,
0-10000
100000
0000-1-2
000001
00-1-200
000100
,
-100000
010000
000010
000001
001000
000100

G:=sub<GL(6,Integers())| [0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,2,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,1,-1,0,0,0,0,2,-1],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-2,1,0,0,-1,0,0,0,0,0,-2,1,0,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C42⋊9D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_9D_4
 
% in TeX
 
G:=Group("C4^2:9D4");
 
// GroupNames label
 
G:=SmallGroup(128,734);
 
// by ID
 
G=gap.SmallGroup(128,734);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,4037]);
 
// 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*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C42⋊9D4 in TeX

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