Copied to
clipboard

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), C248C23, C12.68C24, (C2×C8)⋊29D6, C3⋊C812C23, C87(C22×S3), C62(C2×M4(2)), (S3×C8)⋊21C22, (C2×C24)⋊29C22, (S3×C23).9C4, C4.67(S3×C23), C23.63(C4×S3), C6.31(C23×C4), C8⋊S317C22, C32(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.Dic325C22, (C3×M4(2))⋊29C22, Dic3.27(C22×C4), (C22×Dic3).18C4, (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, C2.32(S3×C22×C4), C22.76(S3×C2×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

C1C6 — C2×S3×M4(2)
C1C3C6C12C4×S3S3×C2×C4S3×C22×C4 — C2×S3×M4(2)
C3C6 — C2×S3×M4(2)
C1C2×C4C2×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 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

Matrix representation of C2×S3×M4(2) in 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] >;

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