Copied to
clipboard

G = 2- 1+4:4S3order 192 = 26·3

1st semidirect product of 2- 1+4 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2- 1+4:4S3, D4:D6:6C2, C4oD4.27D6, (C3xD4).34D4, (C2xC12).22D4, (C3xQ8).34D4, (C2xQ8).96D6, C6.84C22wrC2, C12.219(C2xD4), C3:5(D4.8D4), C12.23D4:8C2, D4.16(C3:D4), (C2xC12).23C23, Q8.23(C3:D4), Q8:3Dic3:13C2, (C6xQ8).99C22, C12.10D4:11C2, (C3x2- 1+4):1C2, C2.18(C24:4S3), (C2xD12).134C22, (C4xDic3).60C22, C4.Dic3.30C22, (C2xC6).46(C2xD4), C4.66(C2xC3:D4), (C2xC4).13(C3:D4), (C2xC4).23(C22xS3), C22.18(C2xC3:D4), (C3xC4oD4).21C22, SmallGroup(192,804)

Series: Derived Chief Lower central Upper central

C1C2xC12 — 2- 1+4:4S3
C1C3C6C2xC6C2xC12C2xD12D4:D6 — 2- 1+4:4S3
C3C6C2xC12 — 2- 1+4:4S3
C1C2C2xC42- 1+4

Generators and relations for 2- 1+4:4S3
 G = < a,b,c,d,e,f | a4=b2=e3=f2=1, c2=d2=a2, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd-1=fcf=a2c, ce=ec, de=ed, fdf=a2cd, fef=e-1 >

Subgroups: 392 in 146 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, M4(2), D8, SD16, C2xD4, C2xQ8, C2xQ8, C4oD4, C4oD4, C3:C8, D12, C2xDic3, C2xC12, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xS3, C4.10D4, C4wrC2, C4.4D4, C8:C22, 2- 1+4, C4.Dic3, C4xDic3, D6:C4, D4:S3, Q8:2S3, C2xD12, C6xQ8, C6xQ8, C3xC4oD4, C3xC4oD4, D4.8D4, C12.10D4, Q8:3Dic3, C12.23D4, D4:D6, C3x2- 1+4, 2- 1+4:4S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C22wrC2, C2xC3:D4, D4.8D4, C24:4S3, 2- 1+4:4S3

Smallest permutation representation of 2- 1+4:4S3
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)
(1 33)(2 36)(3 35)(4 34)(5 31)(6 30)(7 29)(8 32)(9 27)(10 26)(11 25)(12 28)(13 43)(14 42)(15 41)(16 44)(17 39)(18 38)(19 37)(20 40)(21 46)(22 45)(23 48)(24 47)
(1 2 3 4)(5 8 7 6)(9 12 11 10)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 36 35 34)(37 40 39 38)(41 44 43 42)(45 48 47 46)
(1 47 3 45)(2 48 4 46)(5 17 7 19)(6 18 8 20)(9 15 11 13)(10 16 12 14)(21 36 23 34)(22 33 24 35)(25 43 27 41)(26 44 28 42)(29 37 31 39)(30 38 32 40)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 20)(14 19)(15 18)(16 17)(21 23)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 41)(38 44)(39 43)(40 42)(45 48)(46 47)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B···6F8A8B12A···12J
order12222234444444466···68812···12
size11244242224444121224···424244···4

33 irreducible representations

dim11111122222222248
type+++++++++++++
imageC1C2C2C2C2C2S3D4D4D4D6D6C3:D4C3:D4C3:D4D4.8D42- 1+4:4S3
kernel2- 1+4:4S3C12.10D4Q8:3Dic3C12.23D4D4:D6C3x2- 1+42- 1+4C2xC12C3xD4C3xQ8C2xQ8C4oD4C2xC4D4Q8C3C1
# reps11212112221244421

Matrix representation of 2- 1+4:4S3 in GL6(F73)

100000
010000
000100
0072000
0011125
000707072
,
7200000
0720000
0011125
000010
000100
000707072
,
7200000
0720000
000100
0072000
0072727248
003031
,
72710000
010000
0027272718
0000460
0004600
0000046
,
8170000
0640000
001000
000100
000010
000001
,
8170000
65650000
001000
0007200
0072727248
000301

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,1,0,1,70,0,0,0,0,1,70,0,0,0,0,25,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,70,0,0,1,1,0,70,0,0,25,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,3,0,0,1,0,72,0,0,0,0,0,72,3,0,0,0,0,48,1],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,27,0,0,0,0,0,27,0,46,0,0,0,27,46,0,0,0,0,18,0,0,46],[8,0,0,0,0,0,17,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,65,0,0,0,0,17,65,0,0,0,0,0,0,1,0,72,0,0,0,0,72,72,3,0,0,0,0,72,0,0,0,0,0,48,1] >;

2- 1+4:4S3 in GAP, Magma, Sage, TeX

2_-^{1+4}\rtimes_4S_3
% in TeX

G:=Group("ES-(2,2):4S3");
// GroupNames label

G:=SmallGroup(192,804);
// by ID

G=gap.SmallGroup(192,804);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,184,570,1684,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=e^3=f^2=1,c^2=d^2=a^2,b*a*b=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*b,d*c*d^-1=f*c*f=a^2*c,c*e=e*c,d*e=e*d,f*d*f=a^2*c*d,f*e*f=e^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<