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G = C4○D4.D4  order 128 = 27

3rd non-split extension by C4○D4 of D4 acting via D4/C2=C22

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

Aliases: C4○D4.3D4, (C2×D4).70D4, (C22×D4)⋊6C4, C4.58C22≀C2, (C2×Q8).209D4, C23.123(C2×D4), C42⋊C2⋊2C22, C42⋊C22⋊9C2, D4.13(C22⋊C4), C22.12C22≀C2, Q8.13(C22⋊C4), (C22×C4).25C23, C2.20(C24⋊3C4), C23.37D4⋊20C2, C23.31(C22⋊C4), (C2×M4(2))⋊41C22, C23.C23⋊2C2, (C22×D4).10C22, (C2×2+ 1+4).3C2, M4(2).8C22⋊8C2, (C2×C4○D4)⋊5C4, C4.8(C2×C22⋊C4), (C2×C4).227(C2×D4), (C2×D4).203(C2×C4), (C22×C4).15(C2×C4), (C2×Q8).186(C2×C4), (C2×C4○D4).8C22, (C2×C4).44(C22⋊C4), (C2×C4).176(C22×C4), C22.33(C2×C22⋊C4), SmallGroup(128,527)

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

Derived series
C1 — C2×C4 — C4○D4.D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — C4○D4.D4
Lower central
C1 — C2 — C2×C4 — C4○D4.D4
Upper central
C1 — C2 — C22×C4 — C4○D4.D4
Jennings
C1 — C2 — C2 — C22×C4 — C4○D4.D4

Generators and relations for C4○D4.D4
 G = < a,b,c,d,e | a4=c2=d4=1, b2=a2, e2=dad-1=a-1, ab=ba, ac=ca, ae=ea, cbc=ebe-1=a2b, bd=db, dcd-1=ece-1=a2bc, ede-1=a-1d-1 >

Subgroups: 684 in 294 conjugacy classes, 66 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C23⋊C4, C4.D4, C4.10D4, D4⋊C4, C4≀C2, C42⋊C2, C2×M4(2), C22×D4, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, C23.C23, M4(2).8C22, C23.37D4, C42⋊C22, C2×2+ 1+4, C4○D4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C24⋊3C4, C4○D4.D4

Character table of C4○D4.D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11222444444442222444488888888
ρ111111111111111111111111111111    trivial
ρ2111111-1-1111-1-11111-1-1-1-1-11-11-1-111    linear of order 2
ρ3111111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1-1111-1-11111-1-1-1-11-11-111-1-1    linear of order 2
ρ511111-1-11-1-1-1-1111111-11-11111-1-1-1-1    linear of order 2
ρ611111-11-1-1-1-11-11111-11-11-11-1111-1-1    linear of order 2
ρ711111-1-11-1-1-1-1111111-11-1-1-1-1-11111    linear of order 2
ρ811111-11-1-1-1-11-11111-11-111-11-1-1-111    linear of order 2
ρ911-1-1111-1-11-1111-11-11-1-1-1ii-i-ii-i-ii    linear of order 4
ρ1011-1-111-11-11-1-1-11-11-1-1111-iii-i-ii-ii    linear of order 4
ρ1111-1-1111-1-11-1111-11-11-1-1-1-i-iii-iii-i    linear of order 4
ρ1211-1-111-11-11-1-1-11-11-1-1111i-i-iii-ii-i    linear of order 4
ρ1311-1-11-1-1-11-11-111-11-111-11ii-i-i-iii-i    linear of order 4
ρ1411-1-11-1111-111-11-11-1-1-11-1-iii-ii-ii-i    linear of order 4
ρ1511-1-11-1-1-11-11-111-11-111-11-i-iiii-i-ii    linear of order 4
ρ1611-1-11-1111-111-11-11-1-1-11-1i-i-ii-ii-ii    linear of order 4
ρ17222-2-2-20022-200-222-2000000000000    orthogonal lifted from D4
ρ1822-2-2200-20000-2-22-22202000000000    orthogonal lifted from D4
ρ19222220020000-2-2-2-2-220-2000000000    orthogonal lifted from D4
ρ2022-22-20-200002022-2-20-20200000000    orthogonal lifted from D4
ρ2122-22-2-200-22200-2-222000000000000    orthogonal lifted from D4
ρ22222-2-2200-2-2200-222-2000000000000    orthogonal lifted from D4
ρ2322-22-2020000-2022-2-2020-200000000    orthogonal lifted from D4
ρ2422-22-22002-2-200-2-222000000000000    orthogonal lifted from D4
ρ25222-2-2020000-202-2-220-20200000000    orthogonal lifted from D4
ρ26222-2-20-20000202-2-22020-200000000    orthogonal lifted from D4
ρ2722-2-2200200002-22-22-20-2000000000    orthogonal lifted from D4
ρ282222200-200002-2-2-2-2-202000000000    orthogonal lifted from D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of C4○D4.D4
►On 16 points - transitive group 16T232
Generators in S16
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)

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

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

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

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12), (1,13)(2,16)(3,15)(4,10)(5,9)(6,12)(7,11)(8,14), (1,8)(2,7)(3,6)(4,5)(9,12,13,16)(10,15,14,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
 
G:=Group( (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12), (1,13)(2,16)(3,15)(4,10)(5,9)(6,12)(7,11)(8,14), (1,8)(2,7)(3,6)(4,5)(9,12,13,16)(10,15,14,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12)], [(1,7,5,3),(2,4,6,8),(9,11,13,15),(10,16,14,12)], [(1,13),(2,16),(3,15),(4,10),(5,9),(6,12),(7,11),(8,14)], [(1,8),(2,7),(3,6),(4,5),(9,12,13,16),(10,15,14,11)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,232);
 

►On 16 points - transitive group 16T290
Generators in S16
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)

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

(1 5)(2 14)(3 7)(4 16)(6 10)(8 12)

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

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12), (1,9,5,13)(2,14,6,10)(3,11,7,15)(4,16,8,12), (1,5)(2,14)(3,7)(4,16)(6,10)(8,12), (1,14,15,8)(2,7,16,13)(3,12,9,6)(4,5,10,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
 
G:=Group( (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12), (1,9,5,13)(2,14,6,10)(3,11,7,15)(4,16,8,12), (1,5)(2,14)(3,7)(4,16)(6,10)(8,12), (1,14,15,8)(2,7,16,13)(3,12,9,6)(4,5,10,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12)], [(1,9,5,13),(2,14,6,10),(3,11,7,15),(4,16,8,12)], [(1,5),(2,14),(3,7),(4,16),(6,10),(8,12)], [(1,14,15,8),(2,7,16,13),(3,12,9,6),(4,5,10,11)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,290);
 

Matrix representation of C4○D4.D4 ►in GL8(ℤ)

00010000
00100000
0-1000000
-10000000
0000000-1
000000-10
00000100
00001000
,
00100000
00010000
-10000000
0-1000000
00000010
00000001
0000-1000
00000-100
,
-10000000
01000000
00100000
000-10000
000000-10
00000001
0000-1000
00000100
,
00001000
00000100
00000010
00000001
01000000
10000000
00010000
00100000
,
00001000
00000100
000000-10
0000000-1
000-10000
00-100000
0-1000000
-10000000

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

C4○D4.D4 in GAP, Magma, Sage, TeX 

C_4\circ D_4.D_4
 
% in TeX
 
G:=Group("C4oD4.D4");
 
// GroupNames label
 
G:=SmallGroup(128,527);
 
// by ID
 
G=gap.SmallGroup(128,527);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,248,2804,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=c^2=d^4=1,b^2=a^2,e^2=d*a*d^-1=a^-1,a*b=b*a,a*c=c*a,a*e=e*a,c*b*c=e*b*e^-1=a^2*b,b*d=d*b,d*c*d^-1=e*c*e^-1=a^2*b*c,e*d*e^-1=a^-1*d^-1>;
 
// generators/relations
 

Export

Character table of C4○D4.D4 in TeX

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