direct product, metabelian, supersoluble, monomial
Aliases: C3×S3×M4(2), C24⋊22D6, C8⋊6(S3×C6), (S3×C8)⋊7C6, C24⋊7(C2×C6), C8⋊S3⋊5C6, (S3×C24)⋊16C2, (C4×S3).1C12, (S3×C12).4C4, C4.15(S3×C12), D6.9(C2×C12), C3⋊2(C6×M4(2)), C4.Dic3⋊5C6, C12.106(C4×S3), C12.12(C2×C12), (C3×C24)⋊22C22, (C2×C12).321D6, (C3×M4(2))⋊5C6, C22.7(S3×C12), C62.59(C2×C4), (C22×S3).4C12, C6.15(C22×C12), C12.38(C22×C6), Dic3.7(C2×C12), (C6×Dic3).14C4, (C2×Dic3).6C12, C32⋊12(C2×M4(2)), (S3×C12).62C22, (C6×C12).113C22, (C3×C12).170C23, C12.226(C22×S3), (C32×M4(2))⋊7C2, C3⋊C8⋊11(C2×C6), (S3×C2×C6).9C4, (S3×C2×C4).3C6, C4.38(S3×C2×C6), C2.16(S3×C2×C12), C6.114(S3×C2×C4), (C3×C3⋊C8)⋊42C22, (S3×C2×C12).12C2, (C2×C4).45(S3×C6), (C2×C6).62(C4×S3), (C2×C6).5(C2×C12), (C3×C8⋊S3)⋊13C2, (S3×C6).24(C2×C4), (C4×S3).16(C2×C6), (C3×C12).67(C2×C4), (C2×C12).24(C2×C6), (C3×C4.Dic3)⋊21C2, (C3×C6).86(C22×C4), (C3×Dic3).32(C2×C4), SmallGroup(288,677)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×S3×M4(2)
G = < a,b,c,d,e | a3=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: 282 in 146 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C3×S3, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C3×Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), S3×C2×C4, C22×C12, C3×C3⋊C8, C3×C24, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, S3×M4(2), C6×M4(2), S3×C24, C3×C8⋊S3, C3×C4.Dic3, C32×M4(2), S3×C2×C12, C3×S3×M4(2)
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, M4(2), C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C2×M4(2), S3×C6, C3×M4(2), S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, S3×M4(2), C6×M4(2), S3×C2×C12, C3×S3×M4(2)
►On 48 points
(1 30 39)(2 31 40)(3 32 33)(4 25 34)(5 26 35)(6 27 36)(7 28 37)(8 29 38)(9 42 18)(10 43 19)(11 44 20)(12 45 21)(13 46 22)(14 47 23)(15 48 24)(16 41 17)
(1 30 39)(2 31 40)(3 32 33)(4 25 34)(5 26 35)(6 27 36)(7 28 37)(8 29 38)(9 18 42)(10 19 43)(11 20 44)(12 21 45)(13 22 46)(14 23 47)(15 24 48)(16 17 41)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)
G:=sub<Sym(48)| (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,42,18)(10,43,19)(11,44,20)(12,45,21)(13,46,22)(14,47,23)(15,48,24)(16,41,17), (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,18,42)(10,19,43)(11,20,44)(12,21,45)(13,22,46)(14,23,47)(15,24,48)(16,17,41), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)>;
G:=Group( (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,42,18)(10,43,19)(11,44,20)(12,45,21)(13,46,22)(14,47,23)(15,48,24)(16,41,17), (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,18,42)(10,19,43)(11,20,44)(12,21,45)(13,22,46)(14,23,47)(15,24,48)(16,17,41), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47) );
G=PermutationGroup([[(1,30,39),(2,31,40),(3,32,33),(4,25,34),(5,26,35),(6,27,36),(7,28,37),(8,29,38),(9,42,18),(10,43,19),(11,44,20),(12,45,21),(13,46,22),(14,47,23),(15,48,24),(16,41,17)], [(1,30,39),(2,31,40),(3,32,33),(4,25,34),(5,26,35),(6,27,36),(7,28,37),(8,29,38),(9,18,42),(10,19,43),(11,20,44),(12,21,45),(13,22,46),(14,23,47),(15,24,48),(16,17,41)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(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),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | 12N | 12O | 12P | 12Q | 12R | 12S | 12T | 12U | 24A | ··· | 24H | 24I | ··· | 24T | 24U | ··· | 24AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 3 | 3 | 6 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | S3 | D6 | D6 | M4(2) | C3×S3 | C4×S3 | C4×S3 | S3×C6 | S3×C6 | C3×M4(2) | S3×C12 | S3×C12 | S3×M4(2) | C3×S3×M4(2) |
| kernel | C3×S3×M4(2) | S3×C24 | C3×C8⋊S3 | C3×C4.Dic3 | C32×M4(2) | S3×C2×C12 | S3×M4(2) | S3×C12 | C6×Dic3 | S3×C2×C6 | S3×C8 | C8⋊S3 | C4.Dic3 | C3×M4(2) | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | C3×M4(2) | C24 | C2×C12 | C3×S3 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | S3 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 2 | 4 |
Matrix representation of C3×S3×M4(2) ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[0,46,0,0,1,0,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1] >;
C3×S3×M4(2) in GAP, Magma, Sage, TeX
C_3\times S_3\times M_4(2)
% in TeX
G:=Group("C3xS3xM4(2)"); // GroupNames label
G:=SmallGroup(288,677);
// by ID
G=gap.SmallGroup(288,677);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,555,142,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=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