Copied to
clipboard

G = C24.C8order 128 = 27

2nd non-split extension by C24 of C8 acting via C8/C2=C4

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

Aliases: C24.2C8, (C2×C8).15D4, C23.C84C2, (C22×C4).2C8, C23.15(C2×C8), (C23×C4).12C4, C4.37(C23⋊C4), C2.10(C23⋊C8), (C2×M4(2)).6C4, (C2×C4).28M4(2), C4.13(C4.D4), C22.15(C22⋊C8), C24.4C4.10C2, (C2×M4(2)).143C22, (C22×C4).58(C2×C4), (C2×C4).342(C22⋊C4), SmallGroup(128,52)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.C8
C1C2C4C2×C4C2×C8C2×M4(2)C24.4C4 — C24.C8
C1C2C22C23 — C24.C8
C1C4C2×C4C2×M4(2) — C24.C8
C1C2C2C2C2C4C2×C4C2×M4(2) — C24.C8

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

Subgroups: 160 in 64 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C22, C22 [×10], C8 [×3], C2×C4 [×2], C2×C4 [×8], C23, C23 [×4], C16 [×2], C2×C8 [×2], C2×C8, M4(2) [×2], C22×C4 [×2], C22×C4 [×4], C24, C22⋊C8 [×2], M5(2) [×2], C2×M4(2) [×2], C23×C4, C23.C8 [×2], C24.4C4, 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, C24.C8

Character table of C24.C8

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 11244411244444448888888888
ρ111111111111111111111111111    trivial
ρ2111-1-11111-1-111111-1-1-1-1-11-1111    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-1-11111-1-111111-1-1111-11-1-1-1    linear of order 2
ρ5111-1-11111-1-11-1-1-1-111-i-ii-ii-iii    linear of order 4
ρ6111111111111-1-1-1-1-1-1ii-i-i-i-iii    linear of order 4
ρ7111-1-11111-1-11-1-1-1-111ii-ii-ii-i-i    linear of order 4
ρ8111111111111-1-1-1-1-1-1-i-iiiii-i-i    linear of order 4
ρ9111-1-11-1-1-111-1ii-i-ii-iζ85ζ8ζ87ζ83ζ83ζ87ζ85ζ8    linear of order 8
ρ10111111-1-1-1-1-1-1-i-iiii-iζ87ζ83ζ85ζ85ζ8ζ8ζ83ζ87    linear of order 8
ρ11111111-1-1-1-1-1-1-i-iiii-iζ83ζ87ζ8ζ8ζ85ζ85ζ87ζ83    linear of order 8
ρ12111111-1-1-1-1-1-1ii-i-i-iiζ8ζ85ζ83ζ83ζ87ζ87ζ85ζ8    linear of order 8
ρ13111-1-11-1-1-111-1ii-i-ii-iζ8ζ85ζ83ζ87ζ87ζ83ζ8ζ85    linear of order 8
ρ14111-1-11-1-1-111-1-i-iii-iiζ83ζ87ζ8ζ85ζ85ζ8ζ83ζ87    linear of order 8
ρ15111-1-11-1-1-111-1-i-iii-iiζ87ζ83ζ85ζ8ζ8ζ85ζ87ζ83    linear of order 8
ρ16111111-1-1-1-1-1-1ii-i-i-iiζ85ζ8ζ87ζ87ζ83ζ83ζ8ζ85    linear of order 8
ρ1722200-222200-2-22-220000000000    orthogonal lifted from D4
ρ1822200-222200-22-22-20000000000    orthogonal lifted from D4
ρ1922200-2-2-2-20022i-2i-2i2i0000000000    complex lifted from M4(2)
ρ2022200-2-2-2-2002-2i2i2i-2i0000000000    complex lifted from M4(2)
ρ2144-400044-400000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000-4-4400000000000000000    orthogonal lifted from C4.D4
ρ234-40-2204i-4i0-2i2i000000000000000    complex faithful
ρ244-40-220-4i4i02i-2i000000000000000    complex faithful
ρ254-402-20-4i4i0-2i2i000000000000000    complex faithful
ρ264-402-204i-4i02i-2i000000000000000    complex faithful

Permutation representations of C24.C8
On 16 points - transitive group 16T306
Generators in S16
(1 9)(3 11)(4 12)(5 13)(7 15)(8 16)
(2 10)(3 11)(6 14)(7 15)
(2 10)(4 12)(6 14)(8 16)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,306);

Matrix representation of C24.C8 in GL4(𝔽5) generated by

4000
0100
0010
0001
,
4000
0400
0010
0001
,
4000
0100
0040
0001
,
4000
0400
0040
0004
,
0003
1000
0100
0010
G:=sub<GL(4,GF(5))| [4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,1,3,0,0,0] >;

C24.C8 in GAP, Magma, Sage, TeX

C_2^4.C_8
% in TeX

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

G:=SmallGroup(128,52);
// by ID

G=gap.SmallGroup(128,52);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,346,521,136,2804,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=1,e^8=d,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

׿
×
𝔽