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G = C2×S3×M4(2)  order 192 = 26·3

Direct product of C2, S3 and M4(2)

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

Aliases: C2×S3×M4(2), C24⋊8C23, C12.68C24, (C2×C8)⋊29D6, C3⋊C8⋊12C23, C8⋊7(C22×S3), C6⋊2(C2×M4(2)), (S3×C8)⋊21C22, (C2×C24)⋊29C22, (S3×C23).9C4, C4.67(S3×C23), C6.31(C23×C4), C23.63(C4×S3), C8⋊S3⋊17C22, C3⋊2(C22×M4(2)), (C6×M4(2))⋊13C2, (C4×S3).40C23, C12.91(C22×C4), D6.26(C22×C4), (C22×C4).389D6, (C2×C12).881C23, C4.Dic3⋊25C22, (C3×M4(2))⋊29C22, (C22×Dic3).18C4, Dic3.27(C22×C4), (C22×C12).263C22, (S3×C2×C8)⋊28C2, (S3×C2×C4).10C4, C4.122(S3×C2×C4), (C2×C3⋊C8)⋊47C22, (C2×C8⋊S3)⋊26C2, (S3×C22×C4).8C2, C22.76(S3×C2×C4), C2.32(S3×C22×C4), (C4×S3).30(C2×C4), (C2×C4).162(C4×S3), (C2×C12).130(C2×C4), (S3×C2×C4).253C22, (C2×C4.Dic3)⋊24C2, (C2×C6).24(C22×C4), (C22×C6).77(C2×C4), (C22×S3).68(C2×C4), (C2×C4).604(C22×S3), (C2×Dic3).105(C2×C4), SmallGroup(192,1302)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2×S3×M4(2)
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C22×C4 — C2×S3×M4(2)
Lower central
C3 — C6 — C2×S3×M4(2)

Subgroups: 600 in 298 conjugacy classes, 159 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×20], S3 [×4], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], C23, C23 [×10], Dic3 [×4], C12 [×2], C12 [×2], D6 [×8], D6 [×10], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×13], C24, C3⋊C8 [×4], C24 [×4], C4×S3 [×16], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×S3 [×4], C22×S3 [×4], C22×C6, C22×C8 [×2], C2×M4(2), C2×M4(2) [×11], C23×C4, S3×C8 [×8], C8⋊S3 [×8], C2×C3⋊C8 [×2], C4.Dic3 [×4], C2×C24 [×2], C3×M4(2) [×4], S3×C2×C4 [×4], S3×C2×C4 [×8], C22×Dic3, C22×C12, S3×C23, C22×M4(2), S3×C2×C8 [×2], C2×C8⋊S3 [×2], S3×M4(2) [×8], C2×C4.Dic3, C6×M4(2), S3×C22×C4, C2×S3×M4(2)

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], M4(2) [×4], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C2×M4(2) [×6], C23×C4, S3×C2×C4 [×6], S3×C23, C22×M4(2), S3×M4(2) [×2], S3×C22×C4, C2×S3×M4(2)

Generators and relations
 G = < a,b,c,d,e | a2=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

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

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

(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)

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

Matrix representation ►G ⊆ GL4(𝔽73) generated by

72000
07200
00720
00072
,
727200
1000
0010
0001
,
1000
727200
0010
0001
,
72000
07200
004770
004626
,
72000
07200
0010
00772
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,47,46,0,0,70,26],[72,0,0,0,0,72,0,0,0,0,1,7,0,0,0,72] >;
 

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E8A···8H8I···8P12A12B12C12D12E12F24A···24H
order1222222222223444444444444666668···88···812121212121224···24
size1111223333662111122333366222442···26···62222444···4

60 irreducible representations

dim111111111122222224
type+++++++++++
imageC1C2C2C2C2C2C2C4C4C4S3D6D6D6M4(2)C4×S3C4×S3S3×M4(2)
kernelC2×S3×M4(2)S3×C2×C8C2×C8⋊S3S3×M4(2)C2×C4.Dic3C6×M4(2)S3×C22×C4S3×C2×C4C22×Dic3S3×C23C2×M4(2)C2×C8M4(2)C22×C4D6C2×C4C23C2
# reps1228111122212418624

In GAP, Magma, Sage, TeX 

C_2\times S_3\times M_{4(2)}
 
% in TeX
 
G:=Group("C2xS3xM4(2)");
 
// GroupNames label
 
G:=SmallGroup(192,1302);
 
// by ID
 
G=gap.SmallGroup(192,1302);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,80,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
 
// generators/relations
 

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