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G = C8.13C42order 128 = 27

7th non-split extension by C8 of C42 acting via C42/C2×C4=C2

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

Aliases: C8.13C42, C23.5Q16, C8.3(C4⋊C4), (C2×C8).4Q8, (C2×C16).4C4, (C2×C8).83D4, (C2×C4).11D8, C8.C4.4C4, (C2×C4).95SD16, C4.16(C4.Q8), C8.36(C22⋊C4), (C22×C4).191D4, C4.32(D4⋊C4), (C2×M5(2)).13C2, C22.11(C2.D8), C4.6(C2.C42), (C22×C8).204C22, C22.2(Q8⋊C4), C2.15(C22.4Q16), (C2×C8).51(C2×C4), (C2×C4).113(C4⋊C4), (C2×C8.C4).4C2, (C2×C4).64(C22⋊C4), SmallGroup(128,117)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.13C42
C1C2C4C8C2×C8C22×C8C2×M5(2) — C8.13C42
C1C2C4C8 — C8.13C42
C1C4C22×C4C22×C8 — C8.13C42
C1C2C2C2C2C4C4C22×C8 — C8.13C42

Generators and relations for C8.13C42
 G = < a,b,c | a8=1, b4=a4, c4=a6, bab-1=a3, ac=ca, cbc-1=a3b >

Subgroups: 104 in 62 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C22, C8 [×2], C8 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C16 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×6], C22×C4, C8.C4 [×4], C8.C4 [×2], C2×C16 [×2], M5(2) [×2], C22×C8, C2×M4(2) [×2], C2×C8.C4 [×2], C2×M5(2), C8.13C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16, C8.13C42

Smallest permutation representation of C8.13C42
On 32 points
Generators in S32
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)
(1 29 5 25 9 21 13 17)(2 20 14 24 10 28 6 32)(3 27 7 23 11 19 15 31)(4 18 16 22 12 26 8 30)
(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)| (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28), (1,29,5,25,9,21,13,17)(2,20,14,24,10,28,6,32)(3,27,7,23,11,19,15,31)(4,18,16,22,12,26,8,30), (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( (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28), (1,29,5,25,9,21,13,17)(2,20,14,24,10,28,6,32)(3,27,7,23,11,19,15,31)(4,18,16,22,12,26,8,30), (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([(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,23,29,19,25,31,21,27),(18,24,30,20,26,32,22,28)], [(1,29,5,25,9,21,13,17),(2,20,14,24,10,28,6,32),(3,27,7,23,11,19,15,31),(4,18,16,22,12,26,8,30)], [(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)])

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G···8N16A···16H
order12222444448888888···816···16
size11222112222222448···84···4

32 irreducible representations

dim111112222224
type++++-++-
imageC1C2C2C4C4D4Q8D4D8SD16Q16C8.13C42
kernelC8.13C42C2×C8.C4C2×M5(2)C8.C4C2×C16C2×C8C2×C8C22×C4C2×C4C2×C4C23C1
# reps121842112424

Matrix representation of C8.13C42 in GL4(𝔽17) generated by

9000
0900
00150
00015
,
0010
0001
41600
01300
,
8700
13900
00412
001513
G:=sub<GL(4,GF(17))| [9,0,0,0,0,9,0,0,0,0,15,0,0,0,0,15],[0,0,4,0,0,0,16,13,1,0,0,0,0,1,0,0],[8,13,0,0,7,9,0,0,0,0,4,15,0,0,12,13] >;

C8.13C42 in GAP, Magma, Sage, TeX

C_8._{13}C_4^2
% in TeX

G:=Group("C8.13C4^2");
// GroupNames label

G:=SmallGroup(128,117);
// by ID

G=gap.SmallGroup(128,117);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,184,1018,248,1684,242,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^4=a^4,c^4=a^6,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations

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