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G = C8.C16order 128 = 27

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

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: C8.1C16, C16.6Q8, C16.30D4, C42.9C8, M6(2).4C2, C8.22M4(2), C22.5M5(2), C4.8(C2×C16), (C4×C8).28C4, (C2×C8).13C8, C2.5(C4⋊C16), C4.26(C4⋊C8), C8.43(C4⋊C4), (C2×C16).11C4, (C4×C16).17C2, (C2×C16).102C22, (C2×C4).78(C2×C8), (C2×C8).242(C2×C4), SmallGroup(128,154)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.C16
C1C2C4C8C16C2×C16C4×C16 — C8.C16
C1C2C4 — C8.C16
C1C16C2×C16 — C8.C16
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C2×C16 — C8.C16

Generators and relations for C8.C16
 G = < a,b | a8=1, b16=a4, bab-1=a-1 >

2C2
2C4
2C4
2C2×C4
2C16
2C32
2C32

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

56 conjugacy classes

class 1 2A2B4A4B4C···4G8A8B8C8D8E···8J16A···16H16I···16T32A···32P
order122444···488888···816···1616···1632···32
size112112···211112···21···12···24···4

56 irreducible representations

dim1111111122222
type++++-
imageC1C2C2C4C4C8C8C16D4Q8M4(2)M5(2)C8.C16
kernelC8.C16C4×C16M6(2)C4×C8C2×C16C42C2×C8C8C16C16C8C22C1
# reps112224416112416

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

150
08
,
012
10
G:=sub<GL(2,GF(17))| [15,0,0,8],[0,1,12,0] >;

C8.C16 in GAP, Magma, Sage, TeX

C_8.C_{16}
% in TeX

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

G:=SmallGroup(128,154);
// by ID

G=gap.SmallGroup(128,154);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,36,1430,352,80,102,124]);
// Polycyclic

G:=Group<a,b|a^8=1,b^16=a^4,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C8.C16 in TeX

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