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G = C2×S3×C22⋊C4  order 192 = 26·3

Direct product of C2, S3 and C22⋊C4

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

Aliases: C2×S3×C22⋊C4, C24.63D6, C23⋊7(C4×S3), (S3×C23)⋊6C4, (C2×C12)⋊8C23, D6.56(C2×D4), D6⋊6(C22×C4), (C22×C4)⋊38D6, C6.7(C23×C4), D6⋊C4⋊55C22, (S3×C24).2C2, (C2×C6).28C24, C6.33(C22×D4), (C2×Dic3)⋊7C23, C22.123(S3×D4), (C22×C12)⋊33C22, (C22×S3).108D4, C22.17(S3×C23), (C23×C6).54C22, C6.D4⋊44C22, (S3×C23).93C22, C23.155(C22×S3), (C22×C6).120C23, (C22×S3).148C23, (C22×Dic3)⋊40C22, C2.1(C2×S3×D4), C22⋊5(S3×C2×C4), C6⋊1(C2×C22⋊C4), (C2×D6⋊C4)⋊29C2, C2.9(S3×C22×C4), (S3×C2×C4)⋊63C22, (S3×C22×C4)⋊15C2, (C2×C6)⋊2(C22×C4), (C22×C6)⋊8(C2×C4), (C2×C4)⋊7(C22×S3), C3⋊1(C22×C22⋊C4), (C6×C22⋊C4)⋊24C2, (C2×C6).379(C2×D4), (C22×S3)⋊14(C2×C4), (C2×C6.D4)⋊14C2, (C3×C22⋊C4)⋊59C22, SmallGroup(192,1043)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2×S3×C22⋊C4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C24 — C2×S3×C22⋊C4
Lower central
C3 — C6 — C2×S3×C22⋊C4

Subgroups: 1832 in 674 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×16], C3, C4 [×8], C22, C22 [×10], C22 [×88], S3 [×8], S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×28], C23, C23 [×6], C23 [×92], Dic3 [×4], C12 [×4], D6 [×32], D6 [×44], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22⋊C4 [×12], C22×C4 [×2], C22×C4 [×18], C24, C24 [×22], C4×S3 [×16], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], C22×S3 [×36], C22×S3 [×52], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C2×C22⋊C4 [×11], C23×C4 [×2], C25, D6⋊C4 [×8], C6.D4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×8], S3×C2×C4 [×8], C22×Dic3 [×2], C22×C12 [×2], S3×C23 [×2], S3×C23 [×12], S3×C23 [×8], C23×C6, C22×C22⋊C4, S3×C22⋊C4 [×8], C2×D6⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, S3×C22×C4 [×2], S3×C24, C2×S3×C22⋊C4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], D4 [×8], C23 [×15], D6 [×7], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C4×S3 [×4], C22×S3 [×7], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], S3×C2×C4 [×6], S3×D4 [×4], S3×C23, C22×C22⋊C4, S3×C22⋊C4 [×4], S3×C22×C4, C2×S3×D4 [×2], C2×S3×C22⋊C4

Generators and relations
 G = < a,b,c,d,e,f | a2=b3=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
001000
000100
000010
000001
,
100000
010000
00121200
001000
000010
000001
,
1200000
010000
001000
00121200
000010
000001
,
100000
0120000
001000
000100
0000120
0000121
,
100000
010000
001000
000100
0000120
0000012
,
800000
010000
001000
000100
0000111
0000112

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,11,12] >;
 

60 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S2T2U2V2W 3 4A···4H4I···4P6A···6G6H6I6J6K12A···12H
order12···222222···2222234···44···46···6666612···12
size11···122223···3666622···26···62···244444···4

60 irreducible representations

dim111111112222224
type+++++++++++++
imageC1C2C2C2C2C2C2C4S3D4D6D6D6C4×S3S3×D4
kernelC2×S3×C22⋊C4S3×C22⋊C4C2×D6⋊C4C2×C6.D4C6×C22⋊C4S3×C22×C4S3×C24S3×C23C2×C22⋊C4C22×S3C22⋊C4C22×C4C24C23C22
# reps1821121161842184

In GAP, Magma, Sage, TeX 

C_2\times S_3\times C_2^2\rtimes C_4
 
% in TeX
 
G:=Group("C2xS3xC2^2:C4");
 
// GroupNames label
 
G:=SmallGroup(192,1043);
 
// by ID
 
G=gap.SmallGroup(192,1043);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,80,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,e*f=f*e>;
 
// generators/relations
 

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