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G = C8.9C42order 128 = 27

3rd non-split extension by C8 of C42 acting via C42/C2×C4=C2

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

Aliases: C8.9C42, C23.23SD16, (C2×C16)⋊12C4, C8.27(C4⋊C4), (C2×C8).48Q8, C8.C41C4, C2.D8.8C4, (C2×C4).163D8, (C2×C8).361D4, (C2×C4).32Q16, (C22×C16).5C2, (C2×C4).66SD16, C4.22(C2.D8), C8.40(C22⋊C4), (C22×C4).571D4, C4.50(D4⋊C4), C22.9(C4.Q8), C4.14(Q8⋊C4), C2.3(D8.C4), C4.3(C2.C42), (C22×C8).546C22, C22.9(Q8⋊C4), C23.25D4.1C2, C22.45(D4⋊C4), C2.12(C22.4Q16), (C2×C8).168(C2×C4), (C2×C4).110(C4⋊C4), (C2×C8.C4).2C2, (C2×C4).229(C22⋊C4), SmallGroup(128,114)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.9C42
C1C2C4C8C2×C8C22×C8C22×C16 — C8.9C42
C1C2C4C8 — C8.9C42
C1C2×C4C22×C4C22×C8 — C8.9C42
C1C2C2C2C2C4C4C22×C8 — C8.9C42

Generators and relations for C8.9C42
 G = < a,b,c | a8=b4=1, c4=a2, bab-1=a-1, ac=ca, cbc-1=a-1b >

Subgroups: 120 in 66 conjugacy classes, 40 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], 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, C2.D8 [×2], C8.C4 [×2], C8.C4, C2×C16 [×2], C2×C16 [×2], C42⋊C2, C22×C8, C2×M4(2), C23.25D4, C2×C8.C4, C22×C16, C8.9C42
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, D8.C4 [×2], C8.9C42

Smallest permutation representation of C8.9C42
On 64 points
Generators in S64
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)
(1 26 63 42)(2 21 64 37)(3 32 49 48)(4 27 50 43)(5 22 51 38)(6 17 52 33)(7 28 53 44)(8 23 54 39)(9 18 55 34)(10 29 56 45)(11 24 57 40)(12 19 58 35)(13 30 59 46)(14 25 60 41)(15 20 61 36)(16 31 62 47)
(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)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24)(33,43,37,47,41,35,45,39)(34,44,38,48,42,36,46,40)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64), (1,26,63,42)(2,21,64,37)(3,32,49,48)(4,27,50,43)(5,22,51,38)(6,17,52,33)(7,28,53,44)(8,23,54,39)(9,18,55,34)(10,29,56,45)(11,24,57,40)(12,19,58,35)(13,30,59,46)(14,25,60,41)(15,20,61,36)(16,31,62,47), (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( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24)(33,43,37,47,41,35,45,39)(34,44,38,48,42,36,46,40)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64), (1,26,63,42)(2,21,64,37)(3,32,49,48)(4,27,50,43)(5,22,51,38)(6,17,52,33)(7,28,53,44)(8,23,54,39)(9,18,55,34)(10,29,56,45)(11,24,57,40)(12,19,58,35)(13,30,59,46)(14,25,60,41)(15,20,61,36)(16,31,62,47), (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([(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16),(17,27,21,31,25,19,29,23),(18,28,22,32,26,20,30,24),(33,43,37,47,41,35,45,39),(34,44,38,48,42,36,46,40),(49,51,53,55,57,59,61,63),(50,52,54,56,58,60,62,64)], [(1,26,63,42),(2,21,64,37),(3,32,49,48),(4,27,50,43),(5,22,51,38),(6,17,52,33),(7,28,53,44),(8,23,54,39),(9,18,55,34),(10,29,56,45),(11,24,57,40),(12,19,58,35),(13,30,59,46),(14,25,60,41),(15,20,61,36),(16,31,62,47)], [(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 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L16A···16P
order12222244444444448···8888816···16
size11112211112288882···288882···2

44 irreducible representations

dim111111122222222
type+++++-++-
imageC1C2C2C2C4C4C4D4Q8D4D8SD16Q16SD16D8.C4
kernelC8.9C42C23.25D4C2×C8.C4C22×C16C2.D8C8.C4C2×C16C2×C8C2×C8C22×C4C2×C4C2×C4C2×C4C23C2
# reps1111444211222216

Matrix representation of C8.9C42 in GL3(𝔽17) generated by

100
080
01415
,
400
01315
0164
,
400
050
026
G:=sub<GL(3,GF(17))| [1,0,0,0,8,14,0,0,15],[4,0,0,0,13,16,0,15,4],[4,0,0,0,5,2,0,0,6] >;

C8.9C42 in GAP, Magma, Sage, TeX

C_8._9C_4^2
% in TeX

G:=Group("C8.9C4^2");
// GroupNames label

G:=SmallGroup(128,114);
// by ID

G=gap.SmallGroup(128,114);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,520,3924,242,4037,124]);
// Polycyclic

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

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