metabelian, supersoluble, monomial
Aliases: He3⋊7M4(2), C62.5C12, C62.2Dic3, (C6×C12).5S3, (C6×C12).7C6, C12.93(S3×C6), (C3×C12).1C12, C12.58D6⋊C3, C32⋊4C8⋊5C6, C4.(C32⋊C12), (C3×C12).61D6, (C4×He3).4C4, He3⋊3C8⋊12C2, (C3×C12).1Dic3, C6.11(C6×Dic3), C12.3(C3×Dic3), C22.(C32⋊C12), C32⋊2(C3×M4(2)), (C22×He3).5C4, (C4×He3).44C22, C32⋊2(C4.Dic3), (C2×C4×He3).4C2, (C3×C6).6(C2×C12), (C3×C12).16(C2×C6), (C2×C12).23(C3×S3), C4.15(C2×C32⋊C6), C2.3(C2×C32⋊C12), (C3×C6).7(C2×Dic3), C3.2(C3×C4.Dic3), (C2×C4).2(C32⋊C6), (C2×He3).27(C2×C4), (C2×C6).15(C3×Dic3), SmallGroup(432,137)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊7M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d5 >
Subgroups: 257 in 86 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C32, C12, C12, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, He3, C3×C12, C3×C12, C62, C62, C4.Dic3, C3×M4(2), C2×He3, C2×He3, C3×C3⋊C8, C32⋊4C8, C6×C12, C6×C12, C4×He3, C22×He3, C3×C4.Dic3, C12.58D6, He3⋊3C8, C2×C4×He3, He3⋊7M4(2)
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, M4(2), C3×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), C32⋊C6, C6×Dic3, C32⋊C12, C2×C32⋊C6, C3×C4.Dic3, C2×C32⋊C12, He3⋊7M4(2)
►On 72 points
(1 68 47)(2 48 69)(3 70 41)(4 42 71)(5 72 43)(6 44 65)(7 66 45)(8 46 67)(9 22 27)(10 28 23)(11 24 29)(12 30 17)(13 18 31)(14 32 19)(15 20 25)(16 26 21)(33 59 56)(34 49 60)(35 61 50)(36 51 62)(37 63 52)(38 53 64)(39 57 54)(40 55 58)
(1 61 18)(2 19 62)(3 63 20)(4 21 64)(5 57 22)(6 23 58)(7 59 24)(8 17 60)(9 43 39)(10 40 44)(11 45 33)(12 34 46)(13 47 35)(14 36 48)(15 41 37)(16 38 42)(25 70 52)(26 53 71)(27 72 54)(28 55 65)(29 66 56)(30 49 67)(31 68 50)(32 51 69)
(1 47 31)(2 48 32)(3 41 25)(4 42 26)(5 43 27)(6 44 28)(7 45 29)(8 46 30)(9 54 22)(10 55 23)(11 56 24)(12 49 17)(13 50 18)(14 51 19)(15 52 20)(16 53 21)(33 66 59)(34 67 60)(35 68 61)(36 69 62)(37 70 63)(38 71 64)(39 72 57)(40 65 58)
(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)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)
G:=sub<Sym(72)| (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,22,27)(10,28,23)(11,24,29)(12,30,17)(13,18,31)(14,32,19)(15,20,25)(16,26,21)(33,59,56)(34,49,60)(35,61,50)(36,51,62)(37,63,52)(38,53,64)(39,57,54)(40,55,58), (1,61,18)(2,19,62)(3,63,20)(4,21,64)(5,57,22)(6,23,58)(7,59,24)(8,17,60)(9,43,39)(10,40,44)(11,45,33)(12,34,46)(13,47,35)(14,36,48)(15,41,37)(16,38,42)(25,70,52)(26,53,71)(27,72,54)(28,55,65)(29,66,56)(30,49,67)(31,68,50)(32,51,69), (1,47,31)(2,48,32)(3,41,25)(4,42,26)(5,43,27)(6,44,28)(7,45,29)(8,46,30)(9,54,22)(10,55,23)(11,56,24)(12,49,17)(13,50,18)(14,51,19)(15,52,20)(16,53,21)(33,66,59)(34,67,60)(35,68,61)(36,69,62)(37,70,63)(38,71,64)(39,72,57)(40,65,58), (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)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)>;
G:=Group( (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,22,27)(10,28,23)(11,24,29)(12,30,17)(13,18,31)(14,32,19)(15,20,25)(16,26,21)(33,59,56)(34,49,60)(35,61,50)(36,51,62)(37,63,52)(38,53,64)(39,57,54)(40,55,58), (1,61,18)(2,19,62)(3,63,20)(4,21,64)(5,57,22)(6,23,58)(7,59,24)(8,17,60)(9,43,39)(10,40,44)(11,45,33)(12,34,46)(13,47,35)(14,36,48)(15,41,37)(16,38,42)(25,70,52)(26,53,71)(27,72,54)(28,55,65)(29,66,56)(30,49,67)(31,68,50)(32,51,69), (1,47,31)(2,48,32)(3,41,25)(4,42,26)(5,43,27)(6,44,28)(7,45,29)(8,46,30)(9,54,22)(10,55,23)(11,56,24)(12,49,17)(13,50,18)(14,51,19)(15,52,20)(16,53,21)(33,66,59)(34,67,60)(35,68,61)(36,69,62)(37,70,63)(38,71,64)(39,72,57)(40,65,58), (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)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71) );
G=PermutationGroup([[(1,68,47),(2,48,69),(3,70,41),(4,42,71),(5,72,43),(6,44,65),(7,66,45),(8,46,67),(9,22,27),(10,28,23),(11,24,29),(12,30,17),(13,18,31),(14,32,19),(15,20,25),(16,26,21),(33,59,56),(34,49,60),(35,61,50),(36,51,62),(37,63,52),(38,53,64),(39,57,54),(40,55,58)], [(1,61,18),(2,19,62),(3,63,20),(4,21,64),(5,57,22),(6,23,58),(7,59,24),(8,17,60),(9,43,39),(10,40,44),(11,45,33),(12,34,46),(13,47,35),(14,36,48),(15,41,37),(16,38,42),(25,70,52),(26,53,71),(27,72,54),(28,55,65),(29,66,56),(30,49,67),(31,68,50),(32,51,69)], [(1,47,31),(2,48,32),(3,41,25),(4,42,26),(5,43,27),(6,44,28),(7,45,29),(8,46,30),(9,54,22),(10,55,23),(11,56,24),(12,49,17),(13,50,18),(14,51,19),(15,52,20),(16,53,21),(33,66,59),(34,67,60),(35,68,61),(36,69,62),(37,70,63),(38,71,64),(39,72,57),(40,65,58)], [(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),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64),(65,69),(67,71)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6P | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12V | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 18 | ··· | 18 |
62 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 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | Dic3 | M4(2) | C3×S3 | C3×Dic3 | S3×C6 | C3×Dic3 | C4.Dic3 | C3×M4(2) | C3×C4.Dic3 | C32⋊C6 | C32⋊C12 | C2×C32⋊C6 | C32⋊C12 | He3⋊7M4(2) |
| kernel | He3⋊7M4(2) | He3⋊3C8 | C2×C4×He3 | C12.58D6 | C4×He3 | C22×He3 | C32⋊4C8 | C6×C12 | C3×C12 | C62 | C6×C12 | C3×C12 | C3×C12 | C62 | He3 | C2×C12 | C12 | C12 | C2×C6 | C32 | C32 | C3 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 4 |
Matrix representation of He3⋊7M4(2) ►in GL6(𝔽73)
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 72 | 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 7 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 8 | 0 | 56 | 0 | 0 | 0 |
| 72 | 0 | 65 | 0 | 0 | 0 |
| 8 | 64 | 65 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 56 |
| 0 | 0 | 0 | 72 | 0 | 65 |
| 0 | 0 | 0 | 8 | 64 | 65 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,7,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,7,72,72],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[8,72,8,0,0,0,0,0,64,0,0,0,56,65,65,0,0,0,0,0,0,8,72,8,0,0,0,0,0,64,0,0,0,56,65,65],[0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
He3⋊7M4(2) in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_7M_4(2) % in TeX
G:=Group("He3:7M4(2)"); // GroupNames label
G:=SmallGroup(432,137);
// by ID
G=gap.SmallGroup(432,137);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,4037,2035,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations