Copied to
clipboard

G = C165C4order 64 = 26

3rd semidirect product of C16 and C4 acting via C4/C2=C2

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

Aliases: C165C4, C8.2C8, C42.4C4, C4.6C42, C2.1M5(2), C2.3(C4×C8), (C2×C4).2C8, C4.12(C2×C8), C8.22(C2×C4), (C2×C8).12C4, (C2×C16).7C2, (C4×C8).14C2, C22.7(C2×C8), (C2×C8).106C22, (C2×C4).80(C2×C4), SmallGroup(64,27)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C165C4
C1C2C4C2×C4C2×C8C4×C8 — C165C4
C1C2 — C165C4
C1C2×C8 — C165C4
C1C2C2C2C2C4C4C2×C8 — C165C4

Generators and relations for C165C4
 G = < a,b | a16=b4=1, bab-1=a9 >

2C4
2C4

Smallest permutation representation of C165C4
Regular action on 64 points
Generators in S64
(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)
(1 37 64 26)(2 46 49 19)(3 39 50 28)(4 48 51 21)(5 41 52 30)(6 34 53 23)(7 43 54 32)(8 36 55 25)(9 45 56 18)(10 38 57 27)(11 47 58 20)(12 40 59 29)(13 33 60 22)(14 42 61 31)(15 35 62 24)(16 44 63 17)

G:=sub<Sym(64)| (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), (1,37,64,26)(2,46,49,19)(3,39,50,28)(4,48,51,21)(5,41,52,30)(6,34,53,23)(7,43,54,32)(8,36,55,25)(9,45,56,18)(10,38,57,27)(11,47,58,20)(12,40,59,29)(13,33,60,22)(14,42,61,31)(15,35,62,24)(16,44,63,17)>;

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)(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), (1,37,64,26)(2,46,49,19)(3,39,50,28)(4,48,51,21)(5,41,52,30)(6,34,53,23)(7,43,54,32)(8,36,55,25)(9,45,56,18)(10,38,57,27)(11,47,58,20)(12,40,59,29)(13,33,60,22)(14,42,61,31)(15,35,62,24)(16,44,63,17) );

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),(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)], [(1,37,64,26),(2,46,49,19),(3,39,50,28),(4,48,51,21),(5,41,52,30),(6,34,53,23),(7,43,54,32),(8,36,55,25),(9,45,56,18),(10,38,57,27),(11,47,58,20),(12,40,59,29),(13,33,60,22),(14,42,61,31),(15,35,62,24),(16,44,63,17)])

C165C4 is a maximal subgroup of
C16⋊C8  C8.31D8  D8⋊C8  Q16⋊C8  C8.17Q16  C161C8  C4×M5(2)  C162M5(2)  C42.6C8  C8.12M4(2)  C169D4  D8.C8  SD323C4  Q324C4  D164C4  D165C4  C164Q8  C8.12SD16  C8.13SD16  C8.14SD16  C163D4  C8.7D8  C16⋊Q8  C42.4F5
 C16p⋊C4: C32⋊C4  C4810C4  C8017C4  C167F5  C1129C4 ...
 C8p.C8: C16.C8  C24.C8  C40.10C8  C40.C8  C56.C8 ...
C165C4 is a maximal quotient of
C22.7M5(2)
 C16p⋊C4: C32⋊C4  C4810C4  C8017C4  C167F5  C1129C4 ...
 C2p.M5(2): C165C8  C8⋊C16  C24.C8  C40.10C8  C42.4F5  C40.C8  C56.C8 ...

40 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A···8H8I8J8K8L16A···16P
order1222444444448···8888816···16
size1111111122221···122222···2

40 irreducible representations

dim111111112
type+++
imageC1C2C2C4C4C4C8C8M5(2)
kernelC165C4C4×C8C2×C16C16C42C2×C8C8C2×C4C2
# reps112822888

Matrix representation of C165C4 in GL3(𝔽17) generated by

1300
0164
0131
,
1300
001
010
G:=sub<GL(3,GF(17))| [13,0,0,0,16,13,0,4,1],[13,0,0,0,0,1,0,1,0] >;

C165C4 in GAP, Magma, Sage, TeX

C_{16}\rtimes_5C_4
% in TeX

G:=Group("C16:5C4");
// GroupNames label

G:=SmallGroup(64,27);
// by ID

G=gap.SmallGroup(64,27);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,24,409,55,86,88]);
// Polycyclic

G:=Group<a,b|a^16=b^4=1,b*a*b^-1=a^9>;
// generators/relations

Export

Subgroup lattice of C165C4 in TeX

׿
×
𝔽