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G = C24⋊C8order 128 = 27

1st semidirect product of C24 and C8 acting via C8/C2=C4

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

Aliases: C241C8, C25.1C4, C23.14M4(2), C2.1C2≀C4, C23⋊C81C2, (C22×C4).1D4, C2.6(C23⋊C8), C23.13(C2×C8), C24.15(C2×C4), C243C4.2C2, C22.11(C22⋊C8), C22.35(C23⋊C4), C22.6(C4.D4), C23.147(C22⋊C4), (C2×C22⋊C4).3C4, (C2×C22⋊C4).1C22, SmallGroup(128,48)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24⋊C8
C1C2C22C23C22×C4C2×C22⋊C4C243C4 — C24⋊C8
C1C2C22C23 — C24⋊C8
C1C22C23C2×C22⋊C4 — C24⋊C8
C1C22C23C2×C22⋊C4 — C24⋊C8

Generators and relations for C24⋊C8
 G = < a,b,c,d,e | a2=b2=c2=d2=e8=1, ab=ba, ac=ca, ad=da, eae-1=abcd, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 480 in 147 conjugacy classes, 22 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×3], C22 [×3], C22 [×42], C8 [×2], C2×C4 [×7], C23 [×3], C23 [×42], C22⋊C4 [×6], C2×C8 [×2], C22×C4 [×2], C22×C4, C24, C24 [×2], C24 [×8], C22⋊C8 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C25, C23⋊C8 [×2], C243C4, C24⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.D4, C23⋊C8, C2≀C4 [×2], C24⋊C8

Character table of C24⋊C8

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F8A8B8C8D8E8F8G8H
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ2111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-111-1-1-11111-1-1-1-1-11-1111    linear of order 2
ρ4111111-111-1-1-11111-1-1111-11-1-1-1    linear of order 2
ρ5111111111111-1-1-1-1-1-1ii-i-i-i-iii    linear of order 4
ρ6111111111111-1-1-1-1-1-1-i-iiiii-i-i    linear of order 4
ρ7111111-111-1-1-1-1-1-1-111-i-ii-ii-iii    linear of order 4
ρ8111111-111-1-1-1-1-1-1-111ii-ii-ii-i-i    linear of order 4
ρ91-11-11-1-1-1111-1-i-iiii-iζ87ζ83ζ85ζ85ζ8ζ8ζ83ζ87    linear of order 8
ρ101-11-11-1-1-1111-1-i-iiii-iζ83ζ87ζ8ζ8ζ85ζ85ζ87ζ83    linear of order 8
ρ111-11-11-1-1-1111-1ii-i-i-iiζ8ζ85ζ83ζ83ζ87ζ87ζ85ζ8    linear of order 8
ρ121-11-11-1-1-1111-1ii-i-i-iiζ85ζ8ζ87ζ87ζ83ζ83ζ8ζ85    linear of order 8
ρ131-11-11-11-11-1-11ii-i-ii-iζ85ζ8ζ87ζ83ζ83ζ87ζ85ζ8    linear of order 8
ρ141-11-11-11-11-1-11-i-iii-iiζ87ζ83ζ85ζ8ζ8ζ85ζ87ζ83    linear of order 8
ρ151-11-11-11-11-1-11ii-i-ii-iζ8ζ85ζ83ζ87ζ87ζ83ζ8ζ85    linear of order 8
ρ161-11-11-11-11-1-11-i-iii-iiζ83ζ87ζ8ζ85ζ85ζ8ζ83ζ87    linear of order 8
ρ172222220-2-20002-22-20000000000    orthogonal lifted from D4
ρ182222220-2-2000-22-220000000000    orthogonal lifted from D4
ρ192-22-22-202-2000-2i2i2i-2i0000000000    complex lifted from M4(2)
ρ202-22-22-202-20002i-2i-2i2i0000000000    complex lifted from M4(2)
ρ2144-4-400200-22-200000000000000    orthogonal lifted from C2≀C4
ρ2244-4-400-2002-2200000000000000    orthogonal lifted from C2≀C4
ρ234-44-4-4400000000000000000000    orthogonal lifted from C4.D4
ρ244444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ254-4-4400-200-22200000000000000    orthogonal lifted from C2≀C4
ρ264-4-44002002-2-200000000000000    orthogonal lifted from C2≀C4

Permutation representations of C24⋊C8
On 16 points - transitive group 16T228
Generators in S16
(1 10)(4 8)(5 14)(9 13)
(1 14)(2 11)(4 8)(5 10)(6 15)(9 13)
(1 10)(2 6)(3 12)(4 8)(5 14)(7 16)(9 13)(11 15)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

G:=Group( (1,10)(4,8)(5,14)(9,13), (1,14)(2,11)(4,8)(5,10)(6,15)(9,13), (1,10)(2,6)(3,12)(4,8)(5,14)(7,16)(9,13)(11,15), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );

G=PermutationGroup([(1,10),(4,8),(5,14),(9,13)], [(1,14),(2,11),(4,8),(5,10),(6,15),(9,13)], [(1,10),(2,6),(3,12),(4,8),(5,14),(7,16),(9,13),(11,15)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)])

G:=TransitiveGroup(16,228);

On 16 points - transitive group 16T257
Generators in S16
(1 5)(2 15)(3 7)(6 11)(10 14)(12 16)
(1 5)(2 11)(3 12)(4 8)(6 15)(7 16)(9 13)(10 14)
(2 15)(4 9)(6 11)(8 13)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

G:=Group( (1,5)(2,15)(3,7)(6,11)(10,14)(12,16), (1,5)(2,11)(3,12)(4,8)(6,15)(7,16)(9,13)(10,14), (2,15)(4,9)(6,11)(8,13), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );

G=PermutationGroup([(1,5),(2,15),(3,7),(6,11),(10,14),(12,16)], [(1,5),(2,11),(3,12),(4,8),(6,15),(7,16),(9,13),(10,14)], [(2,15),(4,9),(6,11),(8,13)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)])

G:=TransitiveGroup(16,257);

On 16 points - transitive group 16T258
Generators in S16
(1 5)(2 11)(3 7)(4 8)(6 15)(9 13)(10 14)(12 16)
(2 15)(3 16)(6 11)(7 12)
(2 15)(4 9)(6 11)(8 13)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

G:=Group( (1,5)(2,11)(3,7)(4,8)(6,15)(9,13)(10,14)(12,16), (2,15)(3,16)(6,11)(7,12), (2,15)(4,9)(6,11)(8,13), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );

G=PermutationGroup([(1,5),(2,11),(3,7),(4,8),(6,15),(9,13),(10,14),(12,16)], [(2,15),(3,16),(6,11),(7,12)], [(2,15),(4,9),(6,11),(8,13)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)])

G:=TransitiveGroup(16,258);

Matrix representation of C24⋊C8 in GL6(𝔽17)

010000
100000
0016000
001100
0013010
004001
,
1600000
0160000
0016000
001100
0013010
0000016
,
100000
010000
0016000
0001600
0013010
004001
,
100000
010000
0016000
0001600
0000160
0000016
,
020000
1500000
0013020
00001616
0001640
0000130

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,1,13,4,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,13,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,13,4,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,15,0,0,0,0,2,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,2,16,4,13,0,0,0,16,0,0] >;

C24⋊C8 in GAP, Magma, Sage, TeX

C_2^4\rtimes C_8
% in TeX

G:=Group("C2^4:C8");
// GroupNames label

G:=SmallGroup(128,48);
// by ID

G=gap.SmallGroup(128,48);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,346,521,136,2804]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

Export

Character table of C24⋊C8 in TeX

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