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G = C8.8C42order 128 = 27

2nd non-split extension by C8 of C42 acting via C42/C2xC4=C2

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

Aliases: C8.8C42, C23.11Q16, (C2xC16).8C4, C8.26(C4:C4), (C2xC8).52Q8, (C2xC8).353D4, (C2xC4).162D8, C4.9(C4.Q8), C8.C4.3C4, (C22xC16).4C2, (C2xC4).65SD16, C8.34(C22:C4), (C22xC4).570D4, C2.3(C8.4Q8), C4.49(D4:C4), C22.19(C2.D8), C4.2(C2.C42), (C22xC8).545C22, C22.8(Q8:C4), C2.11(C22.4Q16), (C2xC8).178(C2xC4), (C2xC4).109(C4:C4), (C2xC8.C4).1C2, (C2xC4).228(C22:C4), SmallGroup(128,113)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.8C42
C1C2C4C8C2xC8C22xC8C22xC16 — C8.8C42
C1C2C4C8 — C8.8C42
C1C2xC4C22xC4C22xC8 — C8.8C42
C1C2C2C2C2C4C4C22xC8 — 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, C2, C4, C4, C22, C22, C22, C8, C8, C8, C2xC4, C2xC4, C23, C16, C2xC8, C2xC8, C2xC8, M4(2), C22xC4, C8.C4, C8.C4, C2xC16, C2xC16, C22xC8, C2xM4(2), C2xC8.C4, C22xC16, C8.8C42
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C42, C22:C4, C4:C4, D8, SD16, Q16, C2.C42, D4:C4, Q8:C4, C4.Q8, C2.D8, C22.4Q16, C8.4Q8, C8.8C42

Smallest permutation representation of C8.8C42
On 64 points
Generators in S64
(1 24 13 20 9 32 5 28)(2 25 14 21 10 17 6 29)(3 26 15 22 11 18 7 30)(4 27 16 23 12 19 8 31)(33 49 45 61 41 57 37 53)(34 50 46 62 42 58 38 54)(35 51 47 63 43 59 39 55)(36 52 48 64 44 60 40 56)
(1 62 13 50 9 54 5 58)(2 43 14 47 10 35 6 39)(3 60 15 64 11 52 7 56)(4 41 16 45 12 33 8 37)(17 55 29 59 25 63 21 51)(18 36 30 40 26 44 22 48)(19 53 31 57 27 61 23 49)(20 34 32 38 28 42 24 46)
(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)| (1,24,13,20,9,32,5,28)(2,25,14,21,10,17,6,29)(3,26,15,22,11,18,7,30)(4,27,16,23,12,19,8,31)(33,49,45,61,41,57,37,53)(34,50,46,62,42,58,38,54)(35,51,47,63,43,59,39,55)(36,52,48,64,44,60,40,56), (1,62,13,50,9,54,5,58)(2,43,14,47,10,35,6,39)(3,60,15,64,11,52,7,56)(4,41,16,45,12,33,8,37)(17,55,29,59,25,63,21,51)(18,36,30,40,26,44,22,48)(19,53,31,57,27,61,23,49)(20,34,32,38,28,42,24,46), (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( (1,24,13,20,9,32,5,28)(2,25,14,21,10,17,6,29)(3,26,15,22,11,18,7,30)(4,27,16,23,12,19,8,31)(33,49,45,61,41,57,37,53)(34,50,46,62,42,58,38,54)(35,51,47,63,43,59,39,55)(36,52,48,64,44,60,40,56), (1,62,13,50,9,54,5,58)(2,43,14,47,10,35,6,39)(3,60,15,64,11,52,7,56)(4,41,16,45,12,33,8,37)(17,55,29,59,25,63,21,51)(18,36,30,40,26,44,22,48)(19,53,31,57,27,61,23,49)(20,34,32,38,28,42,24,46), (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([[(1,24,13,20,9,32,5,28),(2,25,14,21,10,17,6,29),(3,26,15,22,11,18,7,30),(4,27,16,23,12,19,8,31),(33,49,45,61,41,57,37,53),(34,50,46,62,42,58,38,54),(35,51,47,63,43,59,39,55),(36,52,48,64,44,60,40,56)], [(1,62,13,50,9,54,5,58),(2,43,14,47,10,35,6,39),(3,60,15,64,11,52,7,56),(4,41,16,45,12,33,8,37),(17,55,29,59,25,63,21,51),(18,36,30,40,26,44,22,48),(19,53,31,57,27,61,23,49),(20,34,32,38,28,42,24,46)], [(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
kernelC8.8C42C2xC8.C4C22xC16C8.C4C2xC16C2xC8C2xC8C22xC4C2xC4C2xC4C23C2
# 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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