Copied to
clipboard

## G = C2×S3×M4(2)  order 192 = 26·3

### Direct product of C2, S3 and M4(2)

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)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C2×S3×M4(2)
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 >

Subgroups: 600 in 298 conjugacy classes, 159 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C22×C8, C2×M4(2), C2×M4(2), C23×C4, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C22×M4(2), S3×C2×C8, C2×C8⋊S3, S3×M4(2), C2×C4.Dic3, C6×M4(2), S3×C22×C4, C2×S3×M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C24, C4×S3, C22×S3, C2×M4(2), C23×C4, S3×C2×C4, S3×C23, C22×M4(2), S3×M4(2), S3×C22×C4, C2×S3×M4(2)

Smallest permutation representation of C2×S3×M4(2)
On 48 points
Generators in S48
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 23 46)(2 24 47)(3 17 48)(4 18 41)(5 19 42)(6 20 43)(7 21 44)(8 22 45)(9 35 28)(10 36 29)(11 37 30)(12 38 31)(13 39 32)(14 40 25)(15 33 26)(16 34 27)
(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)
(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)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 8A ··· 8H 8I ··· 8P 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 ··· 8 8 ··· 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 3 3 3 3 6 6 2 1 1 1 1 2 2 3 3 3 3 6 6 2 2 2 4 4 2 ··· 2 6 ··· 6 2 2 2 2 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 D6 M4(2) C4×S3 C4×S3 S3×M4(2) kernel C2×S3×M4(2) S3×C2×C8 C2×C8⋊S3 S3×M4(2) C2×C4.Dic3 C6×M4(2) S3×C22×C4 S3×C2×C4 C22×Dic3 S3×C23 C2×M4(2) C2×C8 M4(2) C22×C4 D6 C2×C4 C23 C2 # reps 1 2 2 8 1 1 1 12 2 2 1 2 4 1 8 6 2 4

Matrix representation of C2×S3×M4(2) in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 72 72 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 47 70 0 0 46 26
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 7 72
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] >;

C2×S3×M4(2) 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

׿
×
𝔽