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), C23.63(C4×S3), C6.31(C23×C4), 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, 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
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)
►On 48 points
(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