direct product, metabelian, supersoluble, monomial
Aliases: M4(2)×C3⋊S3, C24⋊15D6, C24⋊S3⋊9C2, C12.39(C4×S3), C3⋊4(S3×M4(2)), (C3×C24)⋊23C22, (C2×C12).145D6, (C3×M4(2))⋊5S3, C62.63(C2×C4), C32⋊13(C2×M4(2)), C12.58D6⋊11C2, (C3×C12).178C23, (C6×C12).136C22, C12.209(C22×S3), C32⋊4C8⋊30C22, (C32×M4(2))⋊9C2, C8⋊6(C2×C3⋊S3), C6.75(S3×C2×C4), (C8×C3⋊S3)⋊15C2, (C4×C3⋊S3).6C4, C4.15(C4×C3⋊S3), (C2×C6).22(C4×S3), C22.7(C4×C3⋊S3), (C3×C12).73(C2×C4), C4.38(C22×C3⋊S3), (C22×C3⋊S3).13C4, (C4×C3⋊S3).93C22, (C2×C3⋊Dic3).22C4, C3⋊Dic3.49(C2×C4), (C3×C6).106(C22×C4), (C2×C4×C3⋊S3).7C2, C2.16(C2×C4×C3⋊S3), (C2×C4).44(C2×C3⋊S3), (C2×C3⋊S3).43(C2×C4), SmallGroup(288,763)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C3⋊S3
G = < a,b,c,d,e | a8=b2=c3=d3=e2=1, bab=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 660 in 204 conjugacy classes, 75 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C32⋊4C8, C3×C24, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×M4(2), C8×C3⋊S3, C24⋊S3, C12.58D6, C32×M4(2), C2×C4×C3⋊S3, M4(2)×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C3⋊S3, C4×S3, C22×S3, C2×M4(2), C2×C3⋊S3, S3×C2×C4, C4×C3⋊S3, C22×C3⋊S3, S3×M4(2), C2×C4×C3⋊S3, M4(2)×C3⋊S3
►On 72 points
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)
(1 40 18)(2 33 19)(3 34 20)(4 35 21)(5 36 22)(6 37 23)(7 38 24)(8 39 17)(9 59 48)(10 60 41)(11 61 42)(12 62 43)(13 63 44)(14 64 45)(15 57 46)(16 58 47)(25 50 69)(26 51 70)(27 52 71)(28 53 72)(29 54 65)(30 55 66)(31 56 67)(32 49 68)
(1 47 25)(2 48 26)(3 41 27)(4 42 28)(5 43 29)(6 44 30)(7 45 31)(8 46 32)(9 51 33)(10 52 34)(11 53 35)(12 54 36)(13 55 37)(14 56 38)(15 49 39)(16 50 40)(17 57 68)(18 58 69)(19 59 70)(20 60 71)(21 61 72)(22 62 65)(23 63 66)(24 64 67)
(1 5)(2 6)(3 7)(4 8)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)
G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72), (1,40,18)(2,33,19)(3,34,20)(4,35,21)(5,36,22)(6,37,23)(7,38,24)(8,39,17)(9,59,48)(10,60,41)(11,61,42)(12,62,43)(13,63,44)(14,64,45)(15,57,46)(16,58,47)(25,50,69)(26,51,70)(27,52,71)(28,53,72)(29,54,65)(30,55,66)(31,56,67)(32,49,68), (1,47,25)(2,48,26)(3,41,27)(4,42,28)(5,43,29)(6,44,30)(7,45,31)(8,46,32)(9,51,33)(10,52,34)(11,53,35)(12,54,36)(13,55,37)(14,56,38)(15,49,39)(16,50,40)(17,57,68)(18,58,69)(19,59,70)(20,60,71)(21,61,72)(22,62,65)(23,63,66)(24,64,67), (1,5)(2,6)(3,7)(4,8)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)>;
G:=Group( (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72), (1,40,18)(2,33,19)(3,34,20)(4,35,21)(5,36,22)(6,37,23)(7,38,24)(8,39,17)(9,59,48)(10,60,41)(11,61,42)(12,62,43)(13,63,44)(14,64,45)(15,57,46)(16,58,47)(25,50,69)(26,51,70)(27,52,71)(28,53,72)(29,54,65)(30,55,66)(31,56,67)(32,49,68), (1,47,25)(2,48,26)(3,41,27)(4,42,28)(5,43,29)(6,44,30)(7,45,31)(8,46,32)(9,51,33)(10,52,34)(11,53,35)(12,54,36)(13,55,37)(14,56,38)(15,49,39)(16,50,40)(17,57,68)(18,58,69)(19,59,70)(20,60,71)(21,61,72)(22,62,65)(23,63,66)(24,64,67), (1,5)(2,6)(3,7)(4,8)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60) );
G=PermutationGroup([[(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32),(33,37),(35,39),(42,46),(44,48),(49,53),(51,55),(57,61),(59,63),(66,70),(68,72)], [(1,40,18),(2,33,19),(3,34,20),(4,35,21),(5,36,22),(6,37,23),(7,38,24),(8,39,17),(9,59,48),(10,60,41),(11,61,42),(12,62,43),(13,63,44),(14,64,45),(15,57,46),(16,58,47),(25,50,69),(26,51,70),(27,52,71),(28,53,72),(29,54,65),(30,55,66),(31,56,67),(32,49,68)], [(1,47,25),(2,48,26),(3,41,27),(4,42,28),(5,43,29),(6,44,30),(7,45,31),(8,46,32),(9,51,33),(10,52,34),(11,53,35),(12,54,36),(13,55,37),(14,56,38),(15,49,39),(16,50,40),(17,57,68),(18,58,69),(19,59,70),(20,60,71),(21,61,72),(22,62,65),(23,63,66),(24,64,67)], [(1,5),(2,6),(3,7),(4,8),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,65),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,33),(24,34),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,41),(32,42),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 9 | 9 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | S3×M4(2) |
| kernel | M4(2)×C3⋊S3 | C8×C3⋊S3 | C24⋊S3 | C12.58D6 | C32×M4(2) | C2×C4×C3⋊S3 | C4×C3⋊S3 | C2×C3⋊Dic3 | C22×C3⋊S3 | C3×M4(2) | C24 | C2×C12 | C3⋊S3 | C12 | C2×C6 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of M4(2)×C3⋊S3 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
M4(2)×C3⋊S3 in GAP, Magma, Sage, TeX
M_4(2)\times C_3\rtimes S_3
% in TeX
G:=Group("M4(2)xC3:S3"); // GroupNames label
G:=SmallGroup(288,763);
// by ID
G=gap.SmallGroup(288,763);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^2=1,b*a*b=a^5,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations