direct product, metacyclic, nilpotent (class 2), monomial
Aliases: M4(2)×3- 1+2, C72⋊7C6, C36.4C12, C62.3C12, C12.42C62, (C3×C24).7C6, (C2×C36).6C6, (C9×M4(2))⋊C3, C24.15(C3×C6), (C6×C12).11C6, C36.29(C2×C6), C12.9(C3×C12), (C2×C18).3C12, C6.27(C6×C12), (C3×C12).6C12, C9⋊3(C3×M4(2)), C18.14(C2×C12), C32.(C3×M4(2)), C4.(C4×3- 1+2), C8⋊3(C2×3- 1+2), (C32×M4(2)).C3, (C8×3- 1+2)⋊7C2, C3.3(C32×M4(2)), C22.(C4×3- 1+2), (C3×M4(2)).3C32, (C4×3- 1+2).4C4, C4.6(C22×3- 1+2), (C22×3- 1+2).3C4, (C4×3- 1+2).22C22, (C2×C12).15(C3×C6), (C2×C6).11(C3×C12), (C3×C6).35(C2×C12), (C3×C12).69(C2×C6), C2.5(C2×C4×3- 1+2), (C2×C4×3- 1+2).6C2, (C2×C4).2(C2×3- 1+2), (C2×3- 1+2).14(C2×C4), SmallGroup(432,214)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×3- 1+2
G = < a,b,c,d | a8=b2=c9=d3=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 110 in 80 conjugacy classes, 63 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C24, C24, C2×C12, C2×C12, 3- 1+2, C36, C2×C18, C3×C12, C62, C3×M4(2), C3×M4(2), C2×3- 1+2, C2×3- 1+2, C72, C2×C36, C3×C24, C6×C12, C4×3- 1+2, C22×3- 1+2, C9×M4(2), C32×M4(2), C8×3- 1+2, C2×C4×3- 1+2, M4(2)×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C2×C12, 3- 1+2, C3×C12, C62, C3×M4(2), C2×3- 1+2, C6×C12, C4×3- 1+2, C22×3- 1+2, C32×M4(2), C2×C4×3- 1+2, M4(2)×3- 1+2
►On 72 points
(1 66 42 36 53 21 63 17)(2 67 43 28 54 22 55 18)(3 68 44 29 46 23 56 10)(4 69 45 30 47 24 57 11)(5 70 37 31 48 25 58 12)(6 71 38 32 49 26 59 13)(7 72 39 33 50 27 60 14)(8 64 40 34 51 19 61 15)(9 65 41 35 52 20 62 16)
(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 28)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 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 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(46 49 52)(48 54 51)(55 61 58)(56 59 62)(64 70 67)(65 68 71)
G:=sub<Sym(72)| (1,66,42,36,53,21,63,17)(2,67,43,28,54,22,55,18)(3,68,44,29,46,23,56,10)(4,69,45,30,47,24,57,11)(5,70,37,31,48,25,58,12)(6,71,38,32,49,26,59,13)(7,72,39,33,50,27,60,14)(8,64,40,34,51,19,61,15)(9,65,41,35,52,20,62,16), (10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,28)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51)(55,61,58)(56,59,62)(64,70,67)(65,68,71)>;
G:=Group( (1,66,42,36,53,21,63,17)(2,67,43,28,54,22,55,18)(3,68,44,29,46,23,56,10)(4,69,45,30,47,24,57,11)(5,70,37,31,48,25,58,12)(6,71,38,32,49,26,59,13)(7,72,39,33,50,27,60,14)(8,64,40,34,51,19,61,15)(9,65,41,35,52,20,62,16), (10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,28)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51)(55,61,58)(56,59,62)(64,70,67)(65,68,71) );
G=PermutationGroup([[(1,66,42,36,53,21,63,17),(2,67,43,28,54,22,55,18),(3,68,44,29,46,23,56,10),(4,69,45,30,47,24,57,11),(5,70,37,31,48,25,58,12),(6,71,38,32,49,26,59,13),(7,72,39,33,50,27,60,14),(8,64,40,34,51,19,61,15),(9,65,41,35,52,20,62,16)], [(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,28),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(46,49,52),(48,54,51),(55,61,58),(56,59,62),(64,70,67),(65,68,71)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 18A | ··· | 18F | 18G | ··· | 18L | 24A | ··· | 24H | 24I | ··· | 24P | 36A | ··· | 36L | 36M | ··· | 36R | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | C12 | M4(2) | C3×M4(2) | C3×M4(2) | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C4×3- 1+2 | C4×3- 1+2 | M4(2)×3- 1+2 |
| kernel | M4(2)×3- 1+2 | C8×3- 1+2 | C2×C4×3- 1+2 | C9×M4(2) | C32×M4(2) | C4×3- 1+2 | C22×3- 1+2 | C72 | C2×C36 | C3×C24 | C6×C12 | C36 | C2×C18 | C3×C12 | C62 | 3- 1+2 | C9 | C32 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 6 | 2 | 2 | 2 | 12 | 6 | 4 | 2 | 12 | 12 | 4 | 4 | 2 | 12 | 4 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of M4(2)×3- 1+2 ►in GL5(𝔽73)
| 46 | 19 | 0 | 0 | 0 |
| 14 | 27 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 72 |
| 0 | 0 | 0 | 0 | 64 |
| 0 | 0 | 7 | 65 | 9 |
| 64 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 64 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(5,GF(73))| [46,14,0,0,0,19,27,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[1,72,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,8,0,0,0,0,0,64,0,7,0,0,0,0,65,0,0,72,64,9],[64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,72,64,0,0,0,64,0,8] >;
M4(2)×3- 1+2 in GAP, Magma, Sage, TeX
M_4(2)\times 3_-^{1+2} % in TeX
G:=Group("M4(2)xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,214);
// by ID
G=gap.SmallGroup(432,214);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,3053,394,605,242]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^9=d^3=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations