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G = C8.4C42order 128 = 27

4th non-split extension by C8 of C42 acting via C42/C22=C22

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

Aliases: C8.4C42, M5(2)⋊10C4, C23.14SD16, (C2×C8).8Q8, C8.29(C4⋊C4), C8.C44C4, C4.Q8.4C4, (C2×C4).125D8, (C2×C8).355D4, (C2×C4).10Q16, C4.4(C2.D8), (C2×C4).96SD16, C8.38(C22⋊C4), C4.8(Q8⋊C4), (C22×C4).195D4, C4.52(D4⋊C4), C22.4(C4.Q8), (C2×M5(2)).17C2, (C22×C8).208C22, C22.23(D4⋊C4), C2.19(C22.4Q16), C4.10(C2.C42), C22.10(Q8⋊C4), C23.25D4.10C2, (C2×C8).55(C2×C4), (C2×C4).31(C4⋊C4), (C2×C8.C4).7C2, (C2×C4).233(C22⋊C4), SmallGroup(128,121)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.4C42
C1C2C4C8C2×C8C22×C8C2×M5(2) — C8.4C42
C1C2C4C8 — C8.4C42
C1C4C22×C4C22×C8 — C8.4C42
C1C2C2C2C2C4C4C22×C8 — C8.4C42

Generators and relations for C8.4C42
 G = < a,b,c | a8=1, b4=a4, c4=a2, bab-1=a-1, cac-1=a5, cbc-1=a3b >

Subgroups: 120 in 64 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×2], C22 [×3], C22, C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×2], C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×6], C2×C8, M4(2) [×3], C22×C4, C4.Q8 [×2], C2.D8, C8.C4 [×2], C8.C4, C2×C16, M5(2) [×2], M5(2), C42⋊C2, C22×C8, C2×M4(2), C23.25D4, C2×C8.C4, C2×M5(2), C8.4C42
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, C8.4C42

Smallest permutation representation of C8.4C42
On 32 points
Generators in S32
(1 28 5 32 9 20 13 24)(2 21 6 25 10 29 14 17)(3 30 7 18 11 22 15 26)(4 23 8 27 12 31 16 19)
(1 16 30 29 9 8 22 21)(2 28 31 7 10 20 23 15)(3 14 32 27 11 6 24 19)(4 26 17 5 12 18 25 13)
(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)| (1,28,5,32,9,20,13,24)(2,21,6,25,10,29,14,17)(3,30,7,18,11,22,15,26)(4,23,8,27,12,31,16,19), (1,16,30,29,9,8,22,21)(2,28,31,7,10,20,23,15)(3,14,32,27,11,6,24,19)(4,26,17,5,12,18,25,13), (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( (1,28,5,32,9,20,13,24)(2,21,6,25,10,29,14,17)(3,30,7,18,11,22,15,26)(4,23,8,27,12,31,16,19), (1,16,30,29,9,8,22,21)(2,28,31,7,10,20,23,15)(3,14,32,27,11,6,24,19)(4,26,17,5,12,18,25,13), (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([(1,28,5,32,9,20,13,24),(2,21,6,25,10,29,14,17),(3,30,7,18,11,22,15,26),(4,23,8,27,12,31,16,19)], [(1,16,30,29,9,8,22,21),(2,28,31,7,10,20,23,15),(3,14,32,27,11,6,24,19),(4,26,17,5,12,18,25,13)], [(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 2A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J16A···16H
order12222444444444888888888816···16
size1122211222888822224488884···4

32 irreducible representations

dim111111122222224
type+++++-++-
imageC1C2C2C2C4C4C4D4Q8D4D8SD16Q16SD16C8.4C42
kernelC8.4C42C23.25D4C2×C8.C4C2×M5(2)C4.Q8C8.C4M5(2)C2×C8C2×C8C22×C4C2×C4C2×C4C2×C4C23C1
# reps111144421122224

Matrix representation of C8.4C42 in GL4(𝔽17) generated by

121200
51200
0055
00125
,
00013
00130
0100
1000
,
0010
0001
14300
141400
G:=sub<GL(4,GF(17))| [12,5,0,0,12,12,0,0,0,0,5,12,0,0,5,5],[0,0,0,1,0,0,1,0,0,13,0,0,13,0,0,0],[0,0,14,14,0,0,3,14,1,0,0,0,0,1,0,0] >;

C8.4C42 in GAP, Magma, Sage, TeX

C_8._4C_4^2
% in TeX

G:=Group("C8.4C4^2");
// GroupNames label

G:=SmallGroup(128,121);
// by ID

G=gap.SmallGroup(128,121);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,248,1684,242,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^4=a^4,c^4=a^2,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b>;
// generators/relations

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