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G = S3×C4⋊1D4  order 192 = 26·3

Direct product of S3 and C4⋊1D4

direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary

Aliases: S3×C4⋊1D4, C42⋊36D6, C4⋊1(S3×D4), (C4×S3)⋊7D4, C12⋊2(C2×D4), (C2×D4)⋊25D6, Dic3⋊1(C2×D4), D6.61(C2×D4), C4⋊D12⋊16C2, C12⋊3D4⋊25C2, (S3×C42)⋊12C2, (C4×C12)⋊25C22, (C6×D4)⋊17C22, C6.92(C22×D4), (C2×D12)⋊30C22, (C2×C6).258C24, (C2×C12).507C23, (C4×Dic3)⋊65C22, C23.74(C22×S3), (C22×C6).72C23, (S3×C23).71C22, C22.279(S3×C23), (C22×S3).259C23, (C2×Dic3).268C23, (C2×S3×D4)⋊18C2, C3⋊2(C2×C4⋊1D4), C2.65(C2×S3×D4), (C3×C4⋊1D4)⋊5C2, (C2×C3⋊D4)⋊25C22, (S3×C2×C4).250C22, (C2×C4).596(C22×S3), SmallGroup(192,1273)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — S3×C4⋊1D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×D4 — S3×C4⋊1D4
Lower central
C3 — C2×C6 — S3×C4⋊1D4

Subgroups: 1584 in 498 conjugacy classes, 131 normal (12 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×6], C4 [×6], C22, C22 [×46], S3 [×4], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×15], D4 [×48], C23 [×4], C23 [×29], Dic3 [×6], C12 [×6], D6 [×6], D6 [×28], C2×C6, C2×C6 [×12], C42, C42 [×3], C22×C4 [×3], C2×D4 [×6], C2×D4 [×42], C24 [×4], C4×S3 [×12], D12 [×12], C2×Dic3 [×3], C3⋊D4 [×24], C2×C12 [×3], C3×D4 [×12], C22×S3, C22×S3 [×4], C22×S3 [×24], C22×C6 [×4], C2×C42, C4⋊1D4, C4⋊1D4 [×7], C22×D4 [×6], C4×Dic3 [×3], C4×C12, S3×C2×C4 [×3], C2×D12 [×6], S3×D4 [×24], C2×C3⋊D4 [×12], C6×D4 [×6], S3×C23 [×4], C2×C4⋊1D4, S3×C42, C4⋊D12, C12⋊3D4 [×6], C3×C4⋊1D4, C2×S3×D4 [×6], S3×C4⋊1D4

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C22×S3 [×7], C4⋊1D4 [×4], C22×D4 [×3], S3×D4 [×6], S3×C23, C2×C4⋊1D4, C2×S3×D4 [×3], S3×C4⋊1D4

Generators and relations
 G = < a,b,c,d,e | a3=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Smallest permutation representation
►On 48 points
Generators in S48
(1 23 38)(2 24 39)(3 21 40)(4 22 37)(5 29 13)(6 30 14)(7 31 15)(8 32 16)(9 25 44)(10 26 41)(11 27 42)(12 28 43)(17 45 33)(18 46 34)(19 47 35)(20 48 36)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
001000
000100
0000012
0000112
,
1200000
0120000
001000
000100
000001
000010
,
1110000
1120000
001000
000100
0000120
0000012
,
1110000
1120000
0012500
0010100
0000120
0000012
,
1220000
010000
001000
0031200
0000120
0000012

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,2,1,0,0,0,0,0,0,1,3,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A···4F4G···4L6A6B6C6D6E6F6G12A···12F
order122222222222222234···44···4666666612···12
size1111333344441212121222···26···622288884···4

42 irreducible representations

dim11111122224
type+++++++++++
imageC1C2C2C2C2C2S3D4D6D6S3×D4
kernelS3×C4⋊1D4S3×C42C4⋊D12C12⋊3D4C3×C4⋊1D4C2×S3×D4C4⋊1D4C4×S3C42C2×D4C4
# reps111616112166

In GAP, Magma, Sage, TeX 

S_3\times C_4\rtimes_1D_4
 
% in TeX
 
G:=Group("S3xC4:1D4");
 
// GroupNames label
 
G:=SmallGroup(192,1273);
 
// by ID
 
G=gap.SmallGroup(192,1273);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,570,185,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=e^2=1,b*a*b=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,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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