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G = C8.8C42order 128 = 27

2nd non-split extension by C8 of C42 acting via C42/C2×C4=C2

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

Aliases: C8.8C42, C23.11Q16, (C2×C16).8C4, C8.26(C4⋊C4), (C2×C8).52Q8, (C2×C8).353D4, (C2×C4).162D8, C4.9(C4.Q8), C8.C4.3C4, (C22×C16).4C2, (C2×C4).65SD16, C8.34(C22⋊C4), (C22×C4).570D4, C2.3(C8.4Q8), C4.49(D4⋊C4), C22.19(C2.D8), C4.2(C2.C42), (C22×C8).545C22, C22.8(Q8⋊C4), C2.11(C22.4Q16), (C2×C8).178(C2×C4), (C2×C4).109(C4⋊C4), (C2×C8.C4).1C2, (C2×C4).228(C22⋊C4), SmallGroup(128,113)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.8C42
C1C2C4C8C2×C8C22×C8C22×C16 — C8.8C42
C1C2C4C8 — C8.8C42
C1C2×C4C22×C4C22×C8 — C8.8C42
C1C2C2C2C2C4C4C22×C8 — C8.8C42

Generators and relations for C8.8C42
 G = < a,b,c | a8=1, b4=a4, c4=a6, bab-1=a-1, ac=ca, cbc-1=a-1b >

Subgroups: 104 in 64 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], 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], C2×C16 [×2], C22×C8, C2×M4(2) [×2], C2×C8.C4 [×2], C22×C16, C8.8C42
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, C8.4Q8 [×2], C8.8C42

Smallest permutation representation of C8.8C42
On 64 points
Generators in S64
(1 52 13 64 9 60 5 56)(2 53 14 49 10 61 6 57)(3 54 15 50 11 62 7 58)(4 55 16 51 12 63 8 59)(17 33 29 45 25 41 21 37)(18 34 30 46 26 42 22 38)(19 35 31 47 27 43 23 39)(20 36 32 48 28 44 24 40)
(1 44 13 48 9 36 5 40)(2 25 14 29 10 17 6 21)(3 42 15 46 11 34 7 38)(4 23 16 27 12 31 8 19)(18 58 22 54 26 50 30 62)(20 56 24 52 28 64 32 60)(33 61 37 57 41 53 45 49)(35 59 39 55 43 51 47 63)
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)

G:=sub<Sym(64)| (1,52,13,64,9,60,5,56)(2,53,14,49,10,61,6,57)(3,54,15,50,11,62,7,58)(4,55,16,51,12,63,8,59)(17,33,29,45,25,41,21,37)(18,34,30,46,26,42,22,38)(19,35,31,47,27,43,23,39)(20,36,32,48,28,44,24,40), (1,44,13,48,9,36,5,40)(2,25,14,29,10,17,6,21)(3,42,15,46,11,34,7,38)(4,23,16,27,12,31,8,19)(18,58,22,54,26,50,30,62)(20,56,24,52,28,64,32,60)(33,61,37,57,41,53,45,49)(35,59,39,55,43,51,47,63), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)>;

G:=Group( (1,52,13,64,9,60,5,56)(2,53,14,49,10,61,6,57)(3,54,15,50,11,62,7,58)(4,55,16,51,12,63,8,59)(17,33,29,45,25,41,21,37)(18,34,30,46,26,42,22,38)(19,35,31,47,27,43,23,39)(20,36,32,48,28,44,24,40), (1,44,13,48,9,36,5,40)(2,25,14,29,10,17,6,21)(3,42,15,46,11,34,7,38)(4,23,16,27,12,31,8,19)(18,58,22,54,26,50,30,62)(20,56,24,52,28,64,32,60)(33,61,37,57,41,53,45,49)(35,59,39,55,43,51,47,63), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64) );

G=PermutationGroup([(1,52,13,64,9,60,5,56),(2,53,14,49,10,61,6,57),(3,54,15,50,11,62,7,58),(4,55,16,51,12,63,8,59),(17,33,29,45,25,41,21,37),(18,34,30,46,26,42,22,38),(19,35,31,47,27,43,23,39),(20,36,32,48,28,44,24,40)], [(1,44,13,48,9,36,5,40),(2,25,14,29,10,17,6,21),(3,42,15,46,11,34,7,38),(4,23,16,27,12,31,8,19),(18,58,22,54,26,50,30,62),(20,56,24,52,28,64,32,60),(33,61,37,57,41,53,45,49),(35,59,39,55,43,51,47,63)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)])

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F8A···8H8I···8P16A···16P
order1222224444448···88···816···16
size1111221111222···28···82···2

44 irreducible representations

dim111112222222
type++++-++-
imageC1C2C2C4C4D4Q8D4D8SD16Q16C8.4Q8
kernelC8.8C42C2×C8.C4C22×C16C8.C4C2×C16C2×C8C2×C8C22×C4C2×C4C2×C4C23C2
# reps1218421124216

Matrix representation of C8.8C42 in GL3(𝔽17) generated by

100
0913
002
,
1300
074
0810
,
1300
0119
0014
G:=sub<GL(3,GF(17))| [1,0,0,0,9,0,0,13,2],[13,0,0,0,7,8,0,4,10],[13,0,0,0,11,0,0,9,14] >;

C8.8C42 in GAP, Magma, Sage, TeX

C_8._8C_4^2
% in TeX

G:=Group("C8.8C4^2");
// GroupNames label

G:=SmallGroup(128,113);
// by ID

G=gap.SmallGroup(128,113);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,520,248,3924,102,4037,124]);
// Polycyclic

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

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