direct product, metabelian, supersoluble, monomial
Aliases: C3×M4(2)⋊S3, C12.86D12, (C2×D12).6C6, (C6×D12).5C2, C12.55(C3×D4), (C3×C12).40D4, C4.11(C3×D12), (C22×S3).C12, C4.Dic3⋊2C6, C6.49(D6⋊C4), (C2×C12).220D6, M4(2)⋊3(C3×S3), (C3×M4(2))⋊7S3, (C3×M4(2))⋊7C6, C62.37(C2×C4), C22.4(S3×C12), (C6×C12).44C22, C32⋊5(C4.D4), C12.138(C3⋊D4), (C32×M4(2))⋊11C2, (S3×C2×C6).2C4, (C2×C4).1(S3×C6), C2.9(C3×D6⋊C4), (C2×C6).59(C4×S3), (C2×C6).2(C2×C12), C3⋊1(C3×C4.D4), C4.21(C3×C3⋊D4), C6.8(C3×C22⋊C4), (C2×C12).14(C2×C6), (C3×C4.Dic3)⋊18C2, (C3×C6).48(C22⋊C4), SmallGroup(288,257)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M4(2)⋊S3
G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >
Subgroups: 314 in 102 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×6], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×5], M4(2), M4(2), C2×D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8, C24 [×5], D12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4.D4, C3×C12 [×2], S3×C6 [×4], C62, C4.Dic3, C3×M4(2) [×2], C3×M4(2) [×2], C2×D12, C6×D4, C3×C3⋊C8, C3×C24, C3×D12 [×2], C6×C12, S3×C2×C6 [×2], M4(2)⋊S3, C3×C4.D4, C3×C4.Dic3, C32×M4(2), C6×D12, C3×M4(2)⋊S3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C4.D4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, M4(2)⋊S3, C3×C4.D4, C3×D6⋊C4, C3×M4(2)⋊S3
►On 48 points
(1 29 39)(2 30 40)(3 31 33)(4 32 34)(5 25 35)(6 26 36)(7 27 37)(8 28 38)(9 46 22)(10 47 23)(11 48 24)(12 41 17)(13 42 18)(14 43 19)(15 44 20)(16 45 21)
(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)(34 38)(36 40)(41 45)(43 47)
(1 39 29)(2 40 30)(3 33 31)(4 34 32)(5 35 25)(6 36 26)(7 37 27)(8 38 28)(9 46 22)(10 47 23)(11 48 24)(12 41 17)(13 42 18)(14 43 19)(15 44 20)(16 45 21)
(1 13)(2 10)(3 11)(4 16)(5 9)(6 14)(7 15)(8 12)(17 38)(18 39)(19 36)(20 37)(21 34)(22 35)(23 40)(24 33)(25 46)(26 43)(27 44)(28 41)(29 42)(30 47)(31 48)(32 45)
G:=sub<Sym(48)| (1,29,39)(2,30,40)(3,31,33)(4,32,34)(5,25,35)(6,26,36)(7,27,37)(8,28,38)(9,46,22)(10,47,23)(11,48,24)(12,41,17)(13,42,18)(14,43,19)(15,44,20)(16,45,21), (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)(34,38)(36,40)(41,45)(43,47), (1,39,29)(2,40,30)(3,33,31)(4,34,32)(5,35,25)(6,36,26)(7,37,27)(8,38,28)(9,46,22)(10,47,23)(11,48,24)(12,41,17)(13,42,18)(14,43,19)(15,44,20)(16,45,21), (1,13)(2,10)(3,11)(4,16)(5,9)(6,14)(7,15)(8,12)(17,38)(18,39)(19,36)(20,37)(21,34)(22,35)(23,40)(24,33)(25,46)(26,43)(27,44)(28,41)(29,42)(30,47)(31,48)(32,45)>;
G:=Group( (1,29,39)(2,30,40)(3,31,33)(4,32,34)(5,25,35)(6,26,36)(7,27,37)(8,28,38)(9,46,22)(10,47,23)(11,48,24)(12,41,17)(13,42,18)(14,43,19)(15,44,20)(16,45,21), (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)(34,38)(36,40)(41,45)(43,47), (1,39,29)(2,40,30)(3,33,31)(4,34,32)(5,35,25)(6,36,26)(7,37,27)(8,38,28)(9,46,22)(10,47,23)(11,48,24)(12,41,17)(13,42,18)(14,43,19)(15,44,20)(16,45,21), (1,13)(2,10)(3,11)(4,16)(5,9)(6,14)(7,15)(8,12)(17,38)(18,39)(19,36)(20,37)(21,34)(22,35)(23,40)(24,33)(25,46)(26,43)(27,44)(28,41)(29,42)(30,47)(31,48)(32,45) );
G=PermutationGroup([(1,29,39),(2,30,40),(3,31,33),(4,32,34),(5,25,35),(6,26,36),(7,27,37),(8,28,38),(9,46,22),(10,47,23),(11,48,24),(12,41,17),(13,42,18),(14,43,19),(15,44,20),(16,45,21)], [(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),(34,38),(36,40),(41,45),(43,47)], [(1,39,29),(2,40,30),(3,33,31),(4,34,32),(5,35,25),(6,36,26),(7,37,27),(8,38,28),(9,46,22),(10,47,23),(11,48,24),(12,41,17),(13,42,18),(14,43,19),(15,44,20),(16,45,21)], [(1,13),(2,10),(3,11),(4,16),(5,9),(6,14),(7,15),(8,12),(17,38),(18,39),(19,36),(20,37),(21,34),(22,35),(23,40),(24,33),(25,46),(26,43),(27,44),(28,41),(29,42),(30,47),(31,48),(32,45)])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 12A | ··· | 12J | 12K | 12L | 12M | 24A | ··· | 24P | 24Q | 24R | 24S | 24T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | C3×S3 | D12 | C3⋊D4 | C3×D4 | C4×S3 | S3×C6 | C3×D12 | C3×C3⋊D4 | S3×C12 | C4.D4 | M4(2)⋊S3 | C3×C4.D4 | C3×M4(2)⋊S3 |
| kernel | C3×M4(2)⋊S3 | C3×C4.Dic3 | C32×M4(2) | C6×D12 | M4(2)⋊S3 | S3×C2×C6 | C4.Dic3 | C3×M4(2) | C2×D12 | C22×S3 | C3×M4(2) | C3×C12 | C2×C12 | M4(2) | C12 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×M4(2)⋊S3 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 46 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,27,0,0,1,0,0,0,0,0,0,46,0,0,1,0],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;
C3×M4(2)⋊S3 in GAP, Magma, Sage, TeX
C_3\times M_{4(2})\rtimes S_3 % in TeX
G:=Group("C3xM4(2):S3"); // GroupNames label
G:=SmallGroup(288,257);
// by ID
G=gap.SmallGroup(288,257);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,136,1271,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations