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G = 2- 1+44S3order 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+44S3, D4⋊D66C2, C4○D4.27D6, (C3×D4).34D4, (C2×C12).22D4, (C3×Q8).34D4, (C2×Q8).96D6, C6.84C22≀C2, C12.219(C2×D4), C35(D4.8D4), C12.23D48C2, D4.16(C3⋊D4), (C2×C12).23C23, Q8.23(C3⋊D4), Q83Dic313C2, (C6×Q8).99C22, C12.10D411C2, (C3×2- 1+4)⋊1C2, C2.18(C244S3), (C2×D12).134C22, (C4×Dic3).60C22, C4.Dic3.30C22, (C2×C6).46(C2×D4), C4.66(C2×C3⋊D4), (C2×C4).13(C3⋊D4), (C2×C4).23(C22×S3), C22.18(C2×C3⋊D4), (C3×C4○D4).21C22, SmallGroup(192,804)

Series: Derived Chief Lower central Upper central

C1C2×C12 — 2- 1+44S3
C1C3C6C2×C6C2×C12C2×D12D4⋊D6 — 2- 1+44S3
C3C6C2×C12 — 2- 1+44S3
C1C2C2×C42- 1+4

Generators and relations for 2- 1+44S3
 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 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×5], S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], D4 [×2], D4 [×6], Q8 [×2], Q8 [×4], C23, Dic3, C12 [×2], C12 [×4], D6 [×3], C2×C6, C2×C6 [×2], C42, C22⋊C4 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×2], C4○D4 [×4], C3⋊C8 [×2], D12 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×D4 [×4], C3×Q8 [×2], C3×Q8 [×4], C22×S3, C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22 [×2], 2- 1+4, C4.Dic3 [×2], C4×Dic3, D6⋊C4 [×2], D4⋊S3 [×2], Q82S3 [×2], C2×D12, C6×Q8, C6×Q8 [×2], C3×C4○D4 [×2], C3×C4○D4 [×4], D4.8D4, C12.10D4, Q83Dic3 [×2], C12.23D4, D4⋊D6 [×2], C3×2- 1+4, 2- 1+44S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×C3⋊D4 [×3], D4.8D4, C244S3, 2- 1+44S3

Smallest permutation representation of 2- 1+44S3
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+44S3
kernel2- 1+44S3C12.10D4Q83Dic3C12.23D4D4⋊D6C3×2- 1+42- 1+4C2×C12C3×D4C3×Q8C2×Q8C4○D4C2×C4D4Q8C3C1
# reps11212112221244421

Matrix representation of 2- 1+44S3 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+44S3 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

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