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G = S3×C22.D4  order 192 = 26·3

Direct product of S3 and C22.D4

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

Aliases: S3×C22.D4, C4⋊C4⋊27D6, C22⋊C4⋊30D6, D6.44(C2×D4), (C22×C4)⋊40D6, D6⋊C4⋊26C22, (C2×D4).161D6, C22.43(S3×D4), C6.81(C22×D4), D6.39(C4○D4), C23.9D6⋊29C2, D6.D4⋊25C2, (C2×C12).69C23, (C2×C6).196C24, C4⋊Dic3⋊38C22, (C22×S3).96D4, Dic3⋊C4⋊21C22, (C22×C12)⋊38C22, (C6×D4).134C22, C23.35(C22×S3), (C22×C6).31C23, (C2×D12).155C22, C22.D12⋊19C2, C23.28D6⋊20C2, C6.D4⋊28C22, C23.23D6⋊14C2, (S3×C23).56C22, C22.217(S3×C23), (C22×S3).256C23, (C2×Dic3).243C23, (C22×Dic3)⋊45C22, (S3×C4⋊C4)⋊31C2, (C2×S3×D4).8C2, C2.54(C2×S3×D4), (S3×C22×C4)⋊23C2, (S3×C2×C4)⋊70C22, C2.59(S3×C4○D4), (C2×C6).57(C2×D4), (S3×C22⋊C4)⋊10C2, (C3×C4⋊C4)⋊23C22, C6.171(C2×C4○D4), C3⋊4(C2×C22.D4), (C2×C4).60(C22×S3), (C3×C22⋊C4)⋊19C22, (C3×C22.D4)⋊4C2, (C2×C3⋊D4).46C22, SmallGroup(192,1211)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — S3×C22.D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — S3×C22.D4
Lower central
C3 — C2×C6 — S3×C22.D4

Subgroups: 944 in 342 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×10], C22, C22 [×2], C22 [×30], S3 [×4], S3 [×3], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×4], C2×C4 [×23], D4 [×8], C23 [×2], C23 [×17], Dic3 [×5], C12 [×5], D6 [×8], D6 [×17], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×12], C2×D4, C2×D4 [×7], C24 [×2], C4×S3 [×14], D12 [×2], C2×Dic3, C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×4], C2×C12 [×2], C3×D4 [×2], C22×S3 [×3], C22×S3 [×4], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4, C22.D4 [×7], C23×C4, C22×D4, Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4, C6.D4 [×2], C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×6], S3×C2×C4 [×4], C2×D12, S3×D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C22×C12, C6×D4, S3×C23 [×2], C2×C22.D4, S3×C22⋊C4, S3×C22⋊C4 [×2], C23.9D6 [×2], C22.D12, S3×C4⋊C4 [×2], D6.D4 [×2], C23.28D6, C23.23D6, C3×C22.D4, S3×C22×C4, C2×S3×D4, S3×C22.D4

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], S3×D4 [×2], S3×C23, C2×C22.D4, C2×S3×D4, S3×C4○D4 [×2], S3×C22.D4

Generators and relations
 G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=de-1 >

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
001000
000100
0000012
0000112
,
1200000
0120000
0012000
0001200
000001
000010
,
580000
1080000
000800
005000
000010
000001
,
1200000
0120000
0012000
0001200
000010
000001
,
500000
1080000
0001200
0012000
0000120
0000012
,
100000
2120000
001000
0001200
000010
000001

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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,10,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,10,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,2,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N6A6B6C6D6E6F12A12B12C12D12E12F12G
order1222222222222234444444444444466666612121212121212
size1111223333466122222244466661212122224484444888

42 irreducible representations

dim11111111111222222244
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4○D4S3×D4S3×C4○D4
kernelS3×C22.D4S3×C22⋊C4C23.9D6C22.D12S3×C4⋊C4D6.D4C23.28D6C23.23D6C3×C22.D4S3×C22×C4C2×S3×D4C22.D4C22×S3C22⋊C4C4⋊C4C22×C4C2×D4D6C22C2
# reps13212211111143211824

In GAP, Magma, Sage, TeX 

S_3\times C_2^2.D_4
 
% in TeX
 
G:=Group("S3xC2^2.D4");
 
// GroupNames label
 
G:=SmallGroup(192,1211);
 
// by ID
 
G=gap.SmallGroup(192,1211);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,346,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=d*e^-1>;
 
// generators/relations
 

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