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G = C423C8order 128 = 27

3rd semidirect product of C42 and C8 acting via C8/C2=C4

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

Aliases: C423C8, (C2×C42).9C4, (C22×C4).6D4, C429C4.2C2, (C2×C4).30M4(2), C2.2(C42⋊C4), C22.18(C22⋊C8), C22.40(C23⋊C4), C2.2(C42.3C4), C23.152(C22⋊C4), C22.7(C4.10D4), C22.M4(2).3C2, C2.6(C22.M4(2)), (C2×C4⋊C4).7C4, (C2×C4).35(C2×C8), (C2×C4⋊C4).4C22, (C22×C4).61(C2×C4), SmallGroup(128,57)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C423C8
C1C2C22C23C22×C4C2×C4⋊C4C429C4 — C423C8
C1C2C22C2×C4 — C423C8
C1C22C23C2×C4⋊C4 — C423C8
C1C22C23C2×C4⋊C4 — C423C8

Generators and relations for C423C8
 G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=a-1b-1, cbc-1=a2b >

Subgroups: 160 in 67 conjugacy classes, 22 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×9], C22 [×3], C22 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×15], C23, C42 [×2], C42, C4⋊C4 [×6], C2×C8 [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C22⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22.M4(2) [×2], C429C4, C423C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.10D4, C22.M4(2), C42⋊C4, C42.3C4, C423C8

Character table of C423C8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ2111111-11-11111-1-11-1-1-1-11-1111-1    linear of order 2
ρ3111111-11-11111-1-11-1-111-11-1-1-11    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111111-11-1-11-1-1-1-1-111-iii-i-ii-ii    linear of order 4
ρ6111111111-11-1-111-1-1-1i-iii-ii-i-i    linear of order 4
ρ7111111-11-1-11-1-1-1-1-111i-i-iii-ii-i    linear of order 4
ρ8111111111-11-1-111-1-1-1-ii-i-ii-iii    linear of order 4
ρ91-11-11-1-11-1-i-1-ii11ii-iζ87ζ85ζ83ζ83ζ85ζ87ζ8ζ8    linear of order 8
ρ101-11-11-1111i-1i-i-1-1-ii-iζ8ζ83ζ8ζ85ζ87ζ85ζ83ζ87    linear of order 8
ρ111-11-11-1-11-1i-1i-i11-i-iiζ85ζ87ζ8ζ8ζ87ζ85ζ83ζ83    linear of order 8
ρ121-11-11-1-11-1-i-1-ii11ii-iζ83ζ8ζ87ζ87ζ8ζ83ζ85ζ85    linear of order 8
ρ131-11-11-1111-i-1-ii-1-1i-iiζ87ζ85ζ87ζ83ζ8ζ83ζ85ζ8    linear of order 8
ρ141-11-11-1111i-1i-i-1-1-ii-iζ85ζ87ζ85ζ8ζ83ζ8ζ87ζ83    linear of order 8
ρ151-11-11-1-11-1i-1i-i11-i-iiζ8ζ83ζ85ζ85ζ83ζ8ζ87ζ87    linear of order 8
ρ161-11-11-1111-i-1-ii-1-1i-iiζ83ζ8ζ83ζ87ζ85ζ87ζ8ζ85    linear of order 8
ρ172222220-202-2-2-20020000000000    orthogonal lifted from D4
ρ182222220-20-2-22200-20000000000    orthogonal lifted from D4
ρ192-22-22-20-202i2-2i2i00-2i0000000000    complex lifted from M4(2)
ρ202-22-22-20-20-2i22i-2i002i0000000000    complex lifted from M4(2)
ρ214444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ224-4-440020-200002-200000000000    orthogonal lifted from C42⋊C4
ρ234-4-4400-2020000-2200000000000    orthogonal lifted from C42⋊C4
ρ2444-4-40020-20000-2200000000000    symplectic lifted from C42.3C4, Schur index 2
ρ254-44-4-4400000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ2644-4-400-20200002-200000000000    symplectic lifted from C42.3C4, Schur index 2

Smallest permutation representation of C423C8
On 32 points
Generators in S32
(2 23 32 16)(4 10 26 17)(6 19 28 12)(8 14 30 21)
(1 15 31 22)(2 16 32 23)(3 24 25 9)(4 17 26 10)(5 11 27 18)(6 12 28 19)(7 20 29 13)(8 21 30 14)
(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)| (2,23,32,16)(4,10,26,17)(6,19,28,12)(8,14,30,21), (1,15,31,22)(2,16,32,23)(3,24,25,9)(4,17,26,10)(5,11,27,18)(6,12,28,19)(7,20,29,13)(8,21,30,14), (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( (2,23,32,16)(4,10,26,17)(6,19,28,12)(8,14,30,21), (1,15,31,22)(2,16,32,23)(3,24,25,9)(4,17,26,10)(5,11,27,18)(6,12,28,19)(7,20,29,13)(8,21,30,14), (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([(2,23,32,16),(4,10,26,17),(6,19,28,12),(8,14,30,21)], [(1,15,31,22),(2,16,32,23),(3,24,25,9),(4,17,26,10),(5,11,27,18),(6,12,28,19),(7,20,29,13),(8,21,30,14)], [(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 C423C8 in GL6(𝔽17)

100000
16160000
001000
000100
005141014
0076117
,
1600000
0160000
00161000
0010100
0012373
0003610
,
910000
080000
000010
009151315
000100
000002

G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,5,7,0,0,0,1,14,6,0,0,0,0,10,11,0,0,0,0,14,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,10,12,0,0,0,10,1,3,3,0,0,0,0,7,6,0,0,0,0,3,10],[9,0,0,0,0,0,1,8,0,0,0,0,0,0,0,9,0,0,0,0,0,15,1,0,0,0,1,13,0,0,0,0,0,15,0,2] >;

C423C8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3C_8
% in TeX

G:=Group("C4^2:3C8");
// GroupNames label

G:=SmallGroup(128,57);
// by ID

G=gap.SmallGroup(128,57);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,352,1242,521,136,2804]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^2*b>;
// generators/relations

Export

Character table of C423C8 in TeX

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