Copied to
clipboard

G = (C2×C8)⋊4D4  order 128 = 27

4th semidirect product of C2×C8 and D4 acting faithfully

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

Aliases: (C2×C8)⋊4D4, C4.6(C4×D4), C4⋊1D4⋊3C4, (C2×D4).80D4, C4.4D4⋊2C4, C42.8(C2×C4), Q8○M4(2)⋊9C2, (C22×C4).68D4, C4.9C42⋊10C2, C4.100C22≀C2, C23.133(C2×D4), (C22×C4).34C23, C23.37D4⋊25C2, C22.54(C4⋊D4), C23.14(C22⋊C4), C22.29C24.1C2, C23.C23⋊7C2, (C22×D4).32C22, C42⋊C2.32C22, C4.14(C22.D4), C2.49(C23.23D4), (C2×M4(2)).193C22, (C2×D4).89(C2×C4), (C2×C4).245(C2×D4), (C2×Q8).77(C2×C4), (C2×C4).329(C4○D4), (C2×C4).18(C22⋊C4), (C2×C4).194(C22×C4), (C2×C4○D4).28C22, C22.48(C2×C22⋊C4), SmallGroup(128,642)

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

Derived series
C1 — C2×C4 — (C2×C8)⋊4D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — Q8○M4(2) — (C2×C8)⋊4D4
Lower central
C1 — C2 — C2×C4 — (C2×C8)⋊4D4
Upper central
C1 — C2 — C22×C4 — (C2×C8)⋊4D4
Jennings
C1 — C2 — C2 — C22×C4 — (C2×C8)⋊4D4

Generators and relations for (C2×C8)⋊4D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=ab-1, dcd=c-1 >

Subgroups: 412 in 172 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C23⋊C4, D4⋊C4, C42⋊C2, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C2×M4(2), C2×M4(2), C8○D4, C22×D4, C2×C4○D4, C4.9C42, C23.C23, C23.37D4, C22.29C24, Q8○M4(2), (C2×C8)⋊4D4
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, (C2×C8)⋊4D4

Character table of (C2×C8)⋊4D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11222448822224488888844444444
ρ111111111111111111111111111111    trivial
ρ211111-1-1111111-1-1-1-111-1-11-1-1111-1-1    linear of order 2
ρ311111-1-1-1-11111-1-11111-1-1-111-1-1-111    linear of order 2
ρ41111111-1-1111111-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111-1-1111111-1-1-1-1-1-111-111-1-1-111    linear of order 2
ρ611111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ71111111-1-1111111-1-1-1-1-1-111111111    linear of order 2
ρ811111-1-1-1-11111-1-111-1-1111-1-1111-1-1    linear of order 2
ρ911-11-1-1-11-11-11-111-11i-ii-ii-ii-ii-ii-i    linear of order 4
ρ1011-11-1-1-11-11-11-111-11-ii-ii-ii-ii-ii-ii    linear of order 4
ρ1111-11-1-1-1-111-11-1111-1-ii-iii-ii-ii-ii-i    linear of order 4
ρ1211-11-1-1-1-111-11-1111-1i-ii-i-ii-ii-ii-ii    linear of order 4
ρ1311-11-1111-11-11-1-1-11-1i-i-iiii-i-ii-i-ii    linear of order 4
ρ1411-11-1111-11-11-1-1-11-1-iii-i-i-iii-iii-i    linear of order 4
ρ1511-11-111-111-11-1-1-1-11-iii-iii-i-ii-i-ii    linear of order 4
ρ1611-11-111-111-11-1-1-1-11i-i-ii-i-iii-iii-i    linear of order 4
ρ1722-22-22-200-22-22-2200000000000000    orthogonal lifted from D4
ρ1822-2-2200002-2-22000000000-22000-22    orthogonal lifted from D4
ρ1922222-2200-2-2-2-2-2200000000000000    orthogonal lifted from D4
ρ20222222-200-2-2-2-22-200000000000000    orthogonal lifted from D4
ρ2122-22-2-2200-22-222-200000000000000    orthogonal lifted from D4
ρ2222-2-2200002-2-220000000002-20002-2    orthogonal lifted from D4
ρ23222-2-20000-2-222000000002002-2-200    orthogonal lifted from D4
ρ24222-2-20000-2-22200000000-200-22200    orthogonal lifted from D4
ρ25222-2-2000022-2-20000000002i2i000-2i-2i    complex lifted from C4○D4
ρ26222-2-2000022-2-2000000000-2i-2i0002i2i    complex lifted from C4○D4
ρ2722-2-220000-222-200000000-2i002i2i-2i00    complex lifted from C4○D4
ρ2822-2-220000-222-2000000002i00-2i-2i2i00    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of (C2×C8)⋊4D4
►On 16 points - transitive group 16T238
Generators in S16
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)

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

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

(1 10)(2 8)(3 16)(4 6)(5 14)(7 12)(9 11)(13 15)
 
G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,7)(2,11,6,15)(3,5)(4,9,8,13)(10,12)(14,16), (1,10)(2,8)(3,16)(4,6)(5,14)(7,12)(9,11)(13,15)>;
 
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,7)(2,11,6,15)(3,5)(4,9,8,13)(10,12)(14,16), (1,10)(2,8)(3,16)(4,6)(5,14)(7,12)(9,11)(13,15) );
 
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,7),(2,11,6,15),(3,5),(4,9,8,13),(10,12),(14,16)], [(1,10),(2,8),(3,16),(4,6),(5,14),(7,12),(9,11),(13,15)]])
 
G:=TransitiveGroup(16,238);
 

►On 16 points - transitive group 16T298
Generators in S16
(9 13)(10 14)(11 15)(12 16)

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

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

(2 8)(3 7)(4 6)(9 11)(10 14)(13 15)
 
G:=sub<Sym(16)| (9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14,5,10)(2,13)(3,12,7,16)(4,11)(6,9)(8,15), (2,8)(3,7)(4,6)(9,11)(10,14)(13,15)>;
 
G:=Group( (9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14,5,10)(2,13)(3,12,7,16)(4,11)(6,9)(8,15), (2,8)(3,7)(4,6)(9,11)(10,14)(13,15) );
 
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14,5,10),(2,13),(3,12,7,16),(4,11),(6,9),(8,15)], [(2,8),(3,7),(4,6),(9,11),(10,14),(13,15)]])
 
G:=TransitiveGroup(16,298);
 

Matrix representation of (C2×C8)⋊4D4 ►in GL8(ℤ)

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

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

(C2×C8)⋊4D4 in GAP, Magma, Sage, TeX 

(C_2\times C_8)\rtimes_4D_4
 
% in TeX
 
G:=Group("(C2xC8):4D4");
 
// GroupNames label
 
G:=SmallGroup(128,642);
 
// by ID
 
G=gap.SmallGroup(128,642);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,521,248,2804,1411,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,c*b*c^-1=d*b*d=a*b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of (C2×C8)⋊4D4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁