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G = 2+ 1+4.4S3order 192 = 26·3

1st non-split extension by 2+ 1+4 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2+ 1+4.4S3, C4oD4.26D6, (C2xD4).83D6, (C3xD4).33D4, (C3xQ8).33D4, C6.81C22wrC2, Q8.14D6:5C2, C12.218(C2xD4), C3:5(D4.9D4), (C22xC6).25D4, C23.12D6:8C2, C12.D4:12C2, D4.15(C3:D4), (C2xC12).22C23, Q8.22(C3:D4), (C4xDic3):9C22, Q8:3Dic3:12C2, (C2xDic6):16C22, (C6xD4).108C22, C23.13(C3:D4), C4.Dic3:11C22, C2.15(C24:4S3), (C3x2+ 1+4).1C2, (C2xC6).43(C2xD4), C4.65(C2xC3:D4), (C2xC4).22(C22xS3), C22.15(C2xC3:D4), (C3xC4oD4).20C22, SmallGroup(192,801)

Series: Derived Chief Lower central Upper central

C1C2xC12 — 2+ 1+4.4S3
C1C3C6C2xC6C2xC12C2xDic6Q8.14D6 — 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=d2=e3=1, c2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd=fcf-1=a2c, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 392 in 152 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C22:C4, M4(2), SD16, Q16, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C3:C8, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xC6, C22xC6, C4.D4, C4wrC2, C4.4D4, C8.C22, 2+ 1+4, C4.Dic3, C4xDic3, D4.S3, C3:Q16, C6.D4, C2xDic6, C6xD4, C6xD4, C3xC4oD4, C3xC4oD4, D4.9D4, C12.D4, Q8:3Dic3, C23.12D6, Q8.14D6, C3x2+ 1+4, 2+ 1+4.4S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C22wrC2, C2xC3:D4, D4.9D4, 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 45)(2 48)(3 47)(4 46)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)(21 34)(22 33)(23 36)(24 35)(25 43)(26 42)(27 41)(28 44)(29 39)(30 38)(31 37)(32 40)
(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 35)(2 36)(3 33)(4 34)(5 28)(6 25)(7 26)(8 27)(9 32)(10 29)(11 30)(12 31)(13 38)(14 39)(15 40)(16 37)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 22 3 24)(2 21 4 23)(5 40 7 38)(6 39 8 37)(9 42 11 44)(10 41 12 43)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 46 35 48)(34 45 36 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), (1,45)(2,48)(3,47)(4,46)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16)(21,34)(22,33)(23,36)(24,35)(25,43)(26,42)(27,41)(28,44)(29,39)(30,38)(31,37)(32,40), (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), (1,35)(2,36)(3,33)(4,34)(5,28)(6,25)(7,26)(8,27)(9,32)(10,29)(11,30)(12,31)(13,38)(14,39)(15,40)(16,37)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (1,22,3,24)(2,21,4,23)(5,40,7,38)(6,39,8,37)(9,42,11,44)(10,41,12,43)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,46,35,48)(34,45,36,47)>;

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A6B···6J8A8B12A···12F
order12222223444444466···68812···12
size11244442224412122424···424244···4

33 irreducible representations

dim11111122222222248
type++++++++++++-
imageC1C2C2C2C2C2S3D4D4D4D6D6C3:D4C3:D4C3:D4D4.9D42+ 1+4.4S3
kernel2+ 1+4.4S3C12.D4Q8:3Dic3C23.12D6Q8.14D6C3x2+ 1+42+ 1+4C3xD4C3xQ8C22xC6C2xD4C4oD4D4Q8C23C3C1
# reps11212112221244421

Matrix representation of 2+ 1+4.4S3 in GL6(F73)

100000
010000
0046000
0002700
0000460
0000027
,
7200000
0720000
0000027
0000460
0002700
0046000
,
7200000
0720000
0046000
0002700
0000270
0000046
,
010000
100000
0000270
0000046
0046000
0002700
,
36450000
45360000
001000
000100
000010
000001
,
36450000
28370000
0007200
001000
0000046
0000460

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,46,0,0,0,0,27,0,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,27,0,0,0,0,0,0,46,0,0],[36,45,0,0,0,0,45,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,28,0,0,0,0,45,37,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;

2+ 1+4.4S3 in GAP, Magma, Sage, TeX

2_+^{1+4}._4S_3
% in TeX

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

G:=SmallGroup(192,801);
// by ID

G=gap.SmallGroup(192,801);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,570,1684,851,438,102,6278]);
// Polycyclic

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

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