Copied to
clipboard

G = C23.C16order 128 = 27

The non-split extension by C23 of C16 acting via C16/C4=C4

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

Aliases: C23.C16, C16.25D4, M6(2)⋊3C2, C4.7M5(2), C8.21M4(2), (C2×C4).C16, (C2×C8).4C8, (C2×C16).6C4, (C22×C4).5C8, C22.4(C2×C16), (C22×C8).16C4, C8.58(C22⋊C4), C2.7(C22⋊C16), C4.34(C22⋊C8), (C2×C16).49C22, (C2×M5(2)).18C2, (C2×C4).77(C2×C8), (C2×C8).241(C2×C4), SmallGroup(128,132)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.C16
C1C2C4C8C16C2×C16C2×M5(2) — C23.C16
C1C2C22 — C23.C16
C1C8C2×C16 — C23.C16
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C2×C16 — C23.C16

Generators and relations for C23.C16
 G = < a,b,c,d | a2=b2=c2=1, d16=c, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >

2C2
4C2
2C4
2C22
4C22
2C2×C4
2C8
2C2×C4
2C2×C8
2C16
2C2×C8
2C32
2M5(2)
2M5(2)
2C32

Smallest permutation representation of C23.C16
On 32 points
Generators in S32
(2 18)(3 19)(6 22)(7 23)(10 26)(11 27)(14 30)(15 31)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 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)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H16A···16H16I16J16K16L32A···32P
order122244448888888816···161616161632···32
size11241124111122442···244444···4

44 irreducible representations

dim1111111112224
type++++
imageC1C2C2C4C4C8C8C16C16D4M4(2)M5(2)C23.C16
kernelC23.C16M6(2)C2×M5(2)C2×C16C22×C8C2×C8C22×C4C2×C4C23C16C8C4C1
# reps1212244882244

Matrix representation of C23.C16 in GL4(𝔽97) generated by

1000
619600
0010
270096
,
1000
0100
680960
270096
,
96000
09600
00960
00096
,
680950
720361
21290
130700
G:=sub<GL(4,GF(97))| [1,61,0,27,0,96,0,0,0,0,1,0,0,0,0,96],[1,0,68,27,0,1,0,0,0,0,96,0,0,0,0,96],[96,0,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[68,72,2,13,0,0,1,0,95,36,29,70,0,1,0,0] >;

C23.C16 in GAP, Magma, Sage, TeX

C_2^3.C_{16}
% in TeX

G:=Group("C2^3.C16");
// GroupNames label

G:=SmallGroup(128,132);
// by ID

G=gap.SmallGroup(128,132);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,1430,1018,80,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=1,d^16=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

Export

Subgroup lattice of C23.C16 in TeX

׿
×
𝔽