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G = C42⋊D4  order 128 = 27

1st semidirect product of C42 and D4 acting faithfully

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

Aliases: C42⋊1D4, C24.27D4, C4.7(C4×D4), C4⋊1D4⋊4C4, C42⋊1(C2×C4), (C2×D4).81D4, C4.4D4⋊3C4, C4.9C42⋊11C2, C23.134(C2×D4), C4.139(C4⋊D4), C22.11C24⋊4C2, C22.34C22≀C2, (C22×C4).35C23, C42⋊C22⋊16C2, C23.15(C22⋊C4), C22.29C24.2C2, (C22×D4).33C22, C42⋊C2.33C22, C2.50(C23.23D4), (C2×M4(2)).194C22, C22.9(C22.D4), (C2×D4)⋊12(C2×C4), (C2×Q8)⋊12(C2×C4), (C2×C4).246(C2×D4), (C2×C4.D4)⋊20C2, (C2×C4).330(C4○D4), (C2×C4).195(C22×C4), (C2×C4○D4).29C22, C22.49(C2×C22⋊C4), SmallGroup(128,643)

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

Derived series
C1 — C2×C4 — C42⋊D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.11C24 — C42⋊D4
Lower central
C1 — C2 — C2×C4 — C42⋊D4
Upper central
C1 — C2 — C22×C4 — C42⋊D4
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊D4

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

Subgroups: 444 in 181 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4.D4, C4≀C2, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C2×M4(2), C22×D4, C2×C4○D4, C4.9C42, C2×C4.D4, C42⋊C22, C22.11C24, C22.29C24, C42⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C42⋊D4

Character table of C42⋊D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11222444482222444444448888888
ρ111111111111111111111111111111    trivial
ρ211111-1-1-1-1-111111-1-1-1111-11-111-11-1    linear of order 2
ρ311111-1-1-1-1-11111-1111-1-1-111-11-11-11    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ511111-1-1-1-111111-1111-1-1-11-11-11-11-1    linear of order 2
ρ6111111111-11111-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ7111111111-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ811111-1-1-1-1111111-1-1-1111-1-11-1-11-11    linear of order 2
ρ911-11-11-11-1-1-111-1i-ii-i-ii-ii-111ii-i-i    linear of order 4
ρ1011-11-1-11-11-1-111-1-i-ii-ii-iii11-1i-i-ii    linear of order 4
ρ1111-11-11-11-11-111-1-ii-iii-ii-i1-1-1ii-i-i    linear of order 4
ρ1211-11-1-11-111-111-1ii-ii-ii-i-i-1-11i-i-ii    linear of order 4
ρ1311-11-1-11-11-1-111-1ii-ii-ii-i-i11-1-iii-i    linear of order 4
ρ1411-11-11-11-1-1-111-1-ii-iii-ii-i-111-i-iii    linear of order 4
ρ1511-11-1-11-111-111-1-i-ii-ii-iii-1-11-iii-i    linear of order 4
ρ1611-11-11-11-11-111-1i-ii-i-ii-ii1-1-1-i-iii    linear of order 4
ρ1722-2-22000002-22-22000-2-2200000000    orthogonal lifted from D4
ρ1822-22-2-2-22202-2-22000000000000000    orthogonal lifted from D4
ρ1922222-222-20-2-2-2-2000000000000000    orthogonal lifted from D4
ρ2022-2-22000002-22-2-200022-200000000    orthogonal lifted from D4
ρ2122-2-2200000-22-2202-2-200020000000    orthogonal lifted from D4
ρ2222-22-222-2-202-2-22000000000000000    orthogonal lifted from D4
ρ23222222-2-220-2-2-2-2000000000000000    orthogonal lifted from D4
ρ2422-2-2200000-22-220-222000-20000000    orthogonal lifted from D4
ρ25222-2-20000022-2-20-2i-2i2i0002i0000000    complex lifted from C4○D4
ρ26222-2-200000-2-2222i0002i-2i-2i00000000    complex lifted from C4○D4
ρ27222-2-20000022-2-202i2i-2i000-2i0000000    complex lifted from C4○D4
ρ28222-2-200000-2-222-2i000-2i2i2i00000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of C42⋊D4
►On 16 points - transitive group 16T222
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

(2 9)(3 16)(4 5)(6 12)(7 15)(11 13)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,14,10)(2,5,15,11)(3,6,16,12)(4,7,13,9), (1,3,14,16)(2,5)(4,9)(6,8,12,10)(7,13)(11,15), (2,9)(3,16)(4,5)(6,12)(7,15)(11,13)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,14,10)(2,5,15,11)(3,6,16,12)(4,7,13,9), (1,3,14,16)(2,5)(4,9)(6,8,12,10)(7,13)(11,15), (2,9)(3,16)(4,5)(6,12)(7,15)(11,13) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8,14,10),(2,5,15,11),(3,6,16,12),(4,7,13,9)], [(1,3,14,16),(2,5),(4,9),(6,8,12,10),(7,13),(11,15)], [(2,9),(3,16),(4,5),(6,12),(7,15),(11,13)]])
 
G:=TransitiveGroup(16,222);
 

►On 16 points - transitive group 16T276
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

(1 9)(2 11)(3 10)(4 12)(5 13)(6 15)(7 16)(8 14)
 
G:=sub<Sym(16)| (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,15,6,9)(2,13,5,11)(3,16,8,12)(4,14,7,10), (1,9)(2,11)(3,10)(4,12)(5,13)(6,15)(7,16)(8,14)>;
 
G:=Group( (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,15,6,9)(2,13,5,11)(3,16,8,12)(4,14,7,10), (1,9)(2,11)(3,10)(4,12)(5,13)(6,15)(7,16)(8,14) );
 
G=PermutationGroup([[(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,8,6,7),(9,12,11,10),(13,14,15,16)], [(1,15,6,9),(2,13,5,11),(3,16,8,12),(4,14,7,10)], [(1,9),(2,11),(3,10),(4,12),(5,13),(6,15),(7,16),(8,14)]])
 
G:=TransitiveGroup(16,276);
 

►On 16 points - transitive group 16T305
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

(1 16)(2 12)(3 10)(4 14)(5 11)(6 15)(7 13)(8 9)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,4,8)(2,6,3,7)(9,16,11,14)(10,13,12,15), (1,11)(2,13,3,15)(4,9)(5,16)(6,10,7,12)(8,14), (1,16)(2,12)(3,10)(4,14)(5,11)(6,15)(7,13)(8,9)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,4,8)(2,6,3,7)(9,16,11,14)(10,13,12,15), (1,11)(2,13,3,15)(4,9)(5,16)(6,10,7,12)(8,14), (1,16)(2,12)(3,10)(4,14)(5,11)(6,15)(7,13)(8,9) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,5,4,8),(2,6,3,7),(9,16,11,14),(10,13,12,15)], [(1,11),(2,13,3,15),(4,9),(5,16),(6,10,7,12),(8,14)], [(1,16),(2,12),(3,10),(4,14),(5,11),(6,15),(7,13),(8,9)]])
 
G:=TransitiveGroup(16,305);
 

Matrix representation of C42⋊D4 ►in GL8(ℤ)

00001000
00000100
00000010
00000001
00010000
00-100000
0-1000000
10000000
,
01000000
-10000000
00010000
00-100000
00000100
0000-1000
00000001
000000-10
,
01000000
10000000
00010000
00100000
00000001
00000010
00000-100
0000-1000
,
10000000
01000000
00-100000
000-10000
00000010
00000001
00001000
00000100

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

C42⋊D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes D_4
 
% in TeX
 
G:=Group("C4^2:D4");
 
// GroupNames label
 
G:=SmallGroup(128,643);
 
// by ID
 
G=gap.SmallGroup(128,643);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,2804,1411,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C42⋊D4 in TeX

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