Copied to
clipboard

G = M5(2):S3  order 192 = 26·3

5th semidirect product of M5(2) and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.3D8, D24.2C4, C4.11D24, C24.80D4, M5(2):5S3, C8.4(C4xS3), C24.1(C2xC4), (C2xC4).9D12, (C2xC8).46D6, (C2xD24).6C2, (C2xC12).99D4, (C2xC6).8SD16, C4.18(D6:C4), C3:2(M5(2):C2), C8.37(C3:D4), (C3xM5(2)):9C2, C24.C4:11C2, (C2xC24).50C22, C2.9(C2.D24), C6.17(D4:C4), C12.42(C22:C4), C22.6(C24:C2), SmallGroup(192,75)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — M5(2):S3
Chief series
C1 — C3 — C6 — C12 — C24 — C2xC24 — C2xD24 — M5(2):S3
Lower central
C3 — C6 — C12 — C24 — M5(2):S3
Upper central
C1 — C2 — C2xC4 — C2xC8 — M5(2)

Generators and relations for M5(2):S3
 G = < a,b,c,d | a16=b2=c3=d2=1, bab=a9, ac=ca, dad=a11b, bc=cb, bd=db, dcd=c-1 >

Subgroups: 296 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, D4, C23, C12, D6, C2xC6, C16, C2xC8, M4(2), D8, C2xD4, C3:C8, C24, D12, C2xC12, C22xS3, C8.C4, M5(2), C2xD8, C48, D24, D24, C4.Dic3, C2xC24, C2xD12, M5(2):C2, C24.C4, C3xM5(2), C2xD24, M5(2):S3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, D8, SD16, C4xS3, D12, C3:D4, D4:C4, C24:C2, D24, D6:C4, M5(2):C2, C2.D24, M5(2):S3

Smallest permutation representation of M5(2):S3
►On 48 points
Generators in S48
(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)

(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)

(1 36 30)(2 37 31)(3 38 32)(4 39 17)(5 40 18)(6 41 19)(7 42 20)(8 43 21)(9 44 22)(10 45 23)(11 46 24)(12 47 25)(13 48 26)(14 33 27)(15 34 28)(16 35 29)

(2 4)(3 15)(5 13)(6 16)(7 11)(8 14)(10 12)(17 37)(18 48)(19 35)(20 46)(21 33)(22 44)(23 47)(24 42)(25 45)(26 40)(27 43)(28 38)(29 41)(30 36)(31 39)(32 34)
 
G:=sub<Sym(48)| (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47), (1,36,30)(2,37,31)(3,38,32)(4,39,17)(5,40,18)(6,41,19)(7,42,20)(8,43,21)(9,44,22)(10,45,23)(11,46,24)(12,47,25)(13,48,26)(14,33,27)(15,34,28)(16,35,29), (2,4)(3,15)(5,13)(6,16)(7,11)(8,14)(10,12)(17,37)(18,48)(19,35)(20,46)(21,33)(22,44)(23,47)(24,42)(25,45)(26,40)(27,43)(28,38)(29,41)(30,36)(31,39)(32,34)>;
 
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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47), (1,36,30)(2,37,31)(3,38,32)(4,39,17)(5,40,18)(6,41,19)(7,42,20)(8,43,21)(9,44,22)(10,45,23)(11,46,24)(12,47,25)(13,48,26)(14,33,27)(15,34,28)(16,35,29), (2,4)(3,15)(5,13)(6,16)(7,11)(8,14)(10,12)(17,37)(18,48)(19,35)(20,46)(21,33)(22,44)(23,47)(24,42)(25,45)(26,40)(27,43)(28,38)(29,41)(30,36)(31,39)(32,34) );
 
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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(33,41),(35,43),(37,45),(39,47)], [(1,36,30),(2,37,31),(3,38,32),(4,39,17),(5,40,18),(6,41,19),(7,42,20),(8,43,21),(9,44,22),(10,45,23),(11,46,24),(12,47,25),(13,48,26),(14,33,27),(15,34,28),(16,35,29)], [(2,4),(3,15),(5,13),(6,16),(7,11),(8,14),(10,12),(17,37),(18,48),(19,35),(20,46),(21,33),(22,44),(23,47),(24,42),(25,45),(26,40),(27,43),(28,38),(29,41),(30,36),(31,39),(32,34)]])
 

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B6A6B8A8B8C8D8E12A12B12C16A16B16C16D24A24B24C24D24E24F48A···48H
order1222234466888881212121616161624242424242448···48
size112242422224224242422444442222444···4

36 irreducible representations

dim111112222222222244
type+++++++++++++
imageC1C2C2C2C4S3D4D4D6D8SD16C4xS3C3:D4D12D24C24:C2M5(2):C2M5(2):S3
kernelM5(2):S3C24.C4C3xM5(2)C2xD24D24M5(2)C24C2xC12C2xC8C12C2xC6C8C8C2xC4C4C22C3C1
# reps111141111222224424

Matrix representation of M5(2):S3 ►in GL4(F97) generated by

0010
0001
957900
181600
,
1000
0100
00960
00096
,
969600
1000
009696
0010
,
1000
969600
00218
001695
G:=sub<GL(4,GF(97))| [0,0,95,18,0,0,79,16,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,96,0,0,0,0,96],[96,1,0,0,96,0,0,0,0,0,96,1,0,0,96,0],[1,96,0,0,0,96,0,0,0,0,2,16,0,0,18,95] >;
 

M5(2):S3 in GAP, Magma, Sage, TeX 

M_5(2)\rtimes S_3
 
% in TeX
 
G:=Group("M5(2):S3");
 
// GroupNames label
 
G:=SmallGroup(192,75);
 
// by ID
 
G=gap.SmallGroup(192,75);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,387,268,570,1684,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^16=b^2=c^3=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a^11*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<