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G = C16.3C8order 128 = 27

1st non-split extension by C16 of C8 acting via C8/C4=C2

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

Aliases: C16.3C8, C8.37D8, C8.12Q16, C42.47Q8, C8.18M4(2), C4.7(C4⋊C8), C8.16(C2×C8), (C2×C16).13C4, (C4×C16).11C2, (C2×C8).283D4, C2.5(C81C8), C8.C8.4C2, C4.21(C2.D8), (C4×C8).419C22, C22.4(C8.C4), (C2×C8).225(C2×C4), (C2×C4).103(C4⋊C4), SmallGroup(128,105)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C16.3C8
C1C2C4C2×C4C2×C8C4×C8C4×C16 — C16.3C8
C1C2C4C8 — C16.3C8
C1C8C2×C8C4×C8 — C16.3C8
C1C2C2C2C2C2×C4C2×C4C4×C8 — C16.3C8

Generators and relations for C16.3C8
 G = < a,b | a16=1, b8=a8, bab-1=a7 >

2C2
2C4
2C4
2C2×C4
2C16
4C16
4C16
2M5(2)
2M5(2)

Smallest permutation representation of C16.3C8
On 32 points
Generators in S32
(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)
(1 20 11 26 5 32 15 22 9 28 3 18 13 24 7 30)(2 27 12 17 6 23 16 29 10 19 4 25 14 31 8 21)

G:=sub<Sym(32)| (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), (1,20,11,26,5,32,15,22,9,28,3,18,13,24,7,30)(2,27,12,17,6,23,16,29,10,19,4,25,14,31,8,21)>;

G:=Group( (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), (1,20,11,26,5,32,15,22,9,28,3,18,13,24,7,30)(2,27,12,17,6,23,16,29,10,19,4,25,14,31,8,21) );

G=PermutationGroup([(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)], [(1,20,11,26,5,32,15,22,9,28,3,18,13,24,7,30),(2,27,12,17,6,23,16,29,10,19,4,25,14,31,8,21)])

44 conjugacy classes

class 1 2A2B4A4B4C···4G8A8B8C8D8E···8J16A···16P16Q···16X
order122444···488888···816···1616···16
size112112···211112···22···28···8

44 irreducible representations

dim111112222222
type+++-++-
imageC1C2C2C4C8Q8D4M4(2)D8Q16C8.C4C16.3C8
kernelC16.3C8C4×C16C8.C8C2×C16C16C42C2×C8C8C8C8C22C1
# reps1124811222416

Matrix representation of C16.3C8 in GL2(𝔽17) generated by

116
03
,
1410
43
G:=sub<GL(2,GF(17))| [11,0,6,3],[14,4,10,3] >;

C16.3C8 in GAP, Magma, Sage, TeX

C_{16}._3C_8
% in TeX

G:=Group("C16.3C8");
// GroupNames label

G:=SmallGroup(128,105);
// by ID

G=gap.SmallGroup(128,105);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,148,422,436,136,2804,172,124]);
// Polycyclic

G:=Group<a,b|a^16=1,b^8=a^8,b*a*b^-1=a^7>;
// generators/relations

Export

Subgroup lattice of C16.3C8 in TeX

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