Copied to
clipboard

G = C22⋊C16order 64 = 26

The semidirect product of C22 and C16 acting via C16/C8=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C22⋊C16, C8.27D4, C23.2C8, C2.2M5(2), C4.10M4(2), (C2×C16)⋊1C2, (C2×C4).3C8, (C2×C8).6C4, C2.1(C2×C16), (C22×C8).4C2, (C22×C4).8C4, C22.8(C2×C8), C2.2(C22⋊C8), C4.28(C22⋊C4), (C2×C8).107C22, (C2×C4).81(C2×C4), SmallGroup(64,29)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22⋊C16
C1C2C4C8C2×C8C22×C8 — C22⋊C16
C1C2 — C22⋊C16
C1C2×C8 — C22⋊C16
C1C2C2C2C2C4C4C2×C8 — C22⋊C16

Generators and relations for C22⋊C16
 G = < a,b,c | a2=b2=c16=1, cac-1=ab=ba, bc=cb >

2C2
2C2
2C22
2C4
2C22
2C2×C4
2C8
2C2×C4
2C16
2C2×C8
2C2×C8
2C16

Smallest permutation representation of C22⋊C16
On 32 points
Generators in S32
(1 9)(2 29)(3 11)(4 31)(5 13)(6 17)(7 15)(8 19)(10 21)(12 23)(14 25)(16 27)(18 26)(20 28)(22 30)(24 32)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 17)(15 18)(16 19)
(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)| (1,9)(2,29)(3,11)(4,31)(5,13)(6,17)(7,15)(8,19)(10,21)(12,23)(14,25)(16,27)(18,26)(20,28)(22,30)(24,32), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,17)(15,18)(16,19), (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( (1,9)(2,29)(3,11)(4,31)(5,13)(6,17)(7,15)(8,19)(10,21)(12,23)(14,25)(16,27)(18,26)(20,28)(22,30)(24,32), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,17)(15,18)(16,19), (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([(1,9),(2,29),(3,11),(4,31),(5,13),(6,17),(7,15),(8,19),(10,21),(12,23),(14,25),(16,27),(18,26),(20,28),(22,30),(24,32)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,17),(15,18),(16,19)], [(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)])

C22⋊C16 is a maximal subgroup of
C23⋊C16  C23.M4(2)  C22.M5(2)  C23.7M4(2)  C22.SD32  C23.32D8  C23.12SD16  C23.13SD16  C24.5C8  (C2×D4).5C8  C8.12M4(2)  D4×C16  C169D4  C166D4  D87D4  Q167D4  D88D4  D8.9D4  Q16.8D4  D8.10D4  C22.D16  C23.49D8  C23.19D8  C23.50D8  C23.51D8  C23.20D8  A4⋊C16
 C2p.M5(2): C42.13C8  C42.6C8  D6⋊C16  C24.98D4  D101C16  C40.91D4  D10⋊C16  C10.6M5(2) ...
C22⋊C16 is a maximal quotient of
C23⋊C16  C22.M5(2)  Q8⋊C16  C22.7M5(2)  C10.6M5(2)
 D2p⋊C16: D4⋊C16  D6⋊C16  D101C16  D10⋊C16  D14⋊C16 ...
 C8p.D4: C22⋊C32  C23.C16  D4.C16  C24.98D4  C40.91D4  C56.91D4 ...

40 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F8A···8H8I8J8K8L16A···16P
order1222224444448···8888816···16
size1111221111221···122222···2

40 irreducible representations

dim11111111222
type++++
imageC1C2C2C4C4C8C8C16D4M4(2)M5(2)
kernelC22⋊C16C2×C16C22×C8C2×C8C22×C4C2×C4C23C22C8C4C2
# reps121224416224

Matrix representation of C22⋊C16 in GL3(𝔽17) generated by

100
010
0416
,
100
0160
0016
,
300
0415
0013
G:=sub<GL(3,GF(17))| [1,0,0,0,1,4,0,0,16],[1,0,0,0,16,0,0,0,16],[3,0,0,0,4,0,0,15,13] >;

C22⋊C16 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_{16}
% in TeX

G:=Group("C2^2:C16");
// GroupNames label

G:=SmallGroup(64,29);
// by ID

G=gap.SmallGroup(64,29);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,69,88]);
// Polycyclic

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

Export

Subgroup lattice of C22⋊C16 in TeX

׿
×
𝔽