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G = (C2×D4)⋊C8  order 128 = 27

2nd semidirect product of C2×D4 and C8 acting via C8/C2=C4

p-group, metabelian, nilpotent (class 4), monomial

Aliases: (C2×D4)⋊2C8, (C2×C42).5C4, (C22×C4).3D4, C2.8(C23⋊C8), (C22×D4).2C4, (C2×C4).1M4(2), C2.1(C42⋊C4), C22.37(C23⋊C4), C22.13(C22⋊C8), C2.1(C42.C4), C22.8(C4.D4), C23.149(C22⋊C4), C22.M4(2)⋊1C2, C24.3C22.2C2, (C2×C4).1(C2×C8), (C2×C4⋊C4).1C22, (C22×C4).56(C2×C4), SmallGroup(128,50)

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

Derived series
C1 — C2×C4 — (C2×D4)⋊C8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C24.3C22 — (C2×D4)⋊C8
Lower central
C1 — C2 — C22 — C2×C4 — (C2×D4)⋊C8
Upper central
C1 — C22 — C23 — C2×C4⋊C4 — (C2×D4)⋊C8
Jennings
C1 — C22 — C23 — C2×C4⋊C4 — (C2×D4)⋊C8

Generators and relations for (C2×D4)⋊C8
 G = < a,b,c,d | a2=b4=c2=d8=1, ab=ba, ac=ca, dad-1=ab2, cbc=b-1, dbd-1=ab-1, dcd-1=b-1c >

Subgroups: 264 in 85 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C22⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C22.M4(2), C24.3C22, (C2×D4)⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.D4, C23⋊C8, C42⋊C4, C42.C4, (C2×D4)⋊C8

Character table of (C2×D4)⋊C8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11112288444444444488888888
ρ111111111111111111111111111    trivial
ρ2111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-1-1-111111-11-1-1-1-1-11111-1    linear of order 2
ρ4111111-1-1-111111-11-1-1111-1-1-1-11    linear of order 2
ρ511111111-1-11-11-1-1-1-1-1-i-iii-i-iii    linear of order 4
ρ611111111-1-11-11-1-1-1-1-1ii-i-iii-i-i    linear of order 4
ρ7111111-1-11-11-11-11-111ii-ii-i-ii-i    linear of order 4
ρ8111111-1-11-11-11-11-111-i-ii-iii-ii    linear of order 4
ρ91-1-111-1-11-ii1-i-1-i-iiiiζ85ζ8ζ83ζ83ζ8ζ85ζ87ζ87    linear of order 8
ρ101-1-111-11-1-i-i1i-1i-i-iiiζ87ζ83ζ8ζ85ζ87ζ83ζ8ζ85    linear of order 8
ρ111-1-111-11-1-i-i1i-1i-i-iiiζ83ζ87ζ85ζ8ζ83ζ87ζ85ζ8    linear of order 8
ρ121-1-111-1-11-ii1-i-1-i-iiiiζ8ζ85ζ87ζ87ζ85ζ8ζ83ζ83    linear of order 8
ρ131-1-111-1-11i-i1i-1ii-i-i-iζ83ζ87ζ85ζ85ζ87ζ83ζ8ζ8    linear of order 8
ρ141-1-111-11-1ii1-i-1-iii-i-iζ8ζ85ζ87ζ83ζ8ζ85ζ87ζ83    linear of order 8
ρ151-1-111-1-11i-i1i-1ii-i-i-iζ87ζ83ζ8ζ8ζ83ζ87ζ85ζ85    linear of order 8
ρ161-1-111-11-1ii1-i-1-iii-i-iζ85ζ8ζ83ζ87ζ85ζ8ζ83ζ87    linear of order 8
ρ17222222000-2-2-2-22020000000000    orthogonal lifted from D4
ρ182222220002-22-2-20-20000000000    orthogonal lifted from D4
ρ192-2-222-2000-2i-22i2-2i02i0000000000    complex lifted from M4(2)
ρ202-2-222-20002i-2-2i22i0-2i0000000000    complex lifted from M4(2)
ρ214-44-40000-20000020-2200000000    orthogonal lifted from C42⋊C4
ρ224-44-40000200000-202-200000000    orthogonal lifted from C42⋊C4
ρ234444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ244-4-44-4400000000000000000000    orthogonal lifted from C4.D4
ρ2544-4-40000-2i000002i02i-2i00000000    complex lifted from C42.C4
ρ2644-4-400002i00000-2i0-2i2i00000000    complex lifted from C42.C4

Smallest permutation representation of (C2×D4)⋊C8
►On 32 points
Generators in S32
(1 27)(2 6)(3 29)(4 8)(5 31)(7 25)(9 20)(10 14)(11 22)(12 16)(13 24)(15 18)(17 21)(19 23)(26 30)(28 32)

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

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

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

Matrix representation of (C2×D4)⋊C8 ►in GL8(𝔽17)

160000000
016000000
001600000
000160000
000016000
000001600
00000010
00000001
,
001600000
000160000
160000000
016000000
000001600
00001000
00000001
000000160
,
160000000
01000000
001600000
00010000
00001000
000001600
00000001
00000010
,
31014100000
737140000
371470000
10310140000
00000010
00000001
00000100
00001000

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[3,7,3,10,0,0,0,0,10,3,7,3,0,0,0,0,14,7,14,10,0,0,0,0,10,14,7,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

(C2×D4)⋊C8 in GAP, Magma, Sage, TeX 

(C_2\times D_4)\rtimes C_8
 
% in TeX
 
G:=Group("(C2xD4):C8");
 
// GroupNames label
 
G:=SmallGroup(128,50);
 
// by ID
 
G=gap.SmallGroup(128,50);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,723,352,1242,521,136,2804]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^2=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=b^-1*c>;
 
// generators/relations
 

Export

Character table of (C2×D4)⋊C8 in TeX

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