non-abelian, supersoluble, monomial
Aliases: He3⋊5D8, C32⋊4D24, (C3×C24)⋊2S3, (C8×He3)⋊2C2, C24.5(C3⋊S3), He3⋊5D4⋊7C2, (C3×C12).47D6, (C3×C6).24D12, C8⋊1(He3⋊C2), (C2×He3).23D4, C3.2(C32⋊5D8), C2.4(He3⋊5D4), C6.28(C12⋊S3), (C4×He3).36C22, C12.80(C2×C3⋊S3), C4.9(C2×He3⋊C2), SmallGroup(432,176)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊5D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 721 in 121 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3×C6, C24, C24, D12, C3×D4, He3, C3×C12, S3×C6, D24, C3×D8, He3⋊C2, C2×He3, C3×C24, C3×D12, C4×He3, C2×He3⋊C2, C3×D24, C8×He3, He3⋊5D4, He3⋊5D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, D12, C2×C3⋊S3, D24, He3⋊C2, C12⋊S3, C2×He3⋊C2, C32⋊5D8, He3⋊5D4, He3⋊5D8
►On 72 points
Generators in S72
(1 25 47)(2 26 48)(3 27 41)(4 28 42)(5 29 43)(6 30 44)(7 31 45)(8 32 46)(9 33 51)(10 34 52)(11 35 53)(12 36 54)(13 37 55)(14 38 56)(15 39 49)(16 40 50)(17 68 57)(18 69 58)(19 70 59)(20 71 60)(21 72 61)(22 65 62)(23 66 63)(24 67 64)
(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 69 58)(2 70 59)(3 71 60)(4 72 61)(5 65 62)(6 66 63)(7 67 64)(8 68 57)(9 19 51)(10 20 52)(11 21 53)(12 22 54)(13 23 55)(14 24 56)(15 17 49)(16 18 50)(25 47 40)(26 48 33)(27 41 34)(28 42 35)(29 43 36)(30 44 37)(31 45 38)(32 46 39)
(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)
(1 8)(2 7)(3 6)(4 5)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 18)(19 24)(20 23)(21 22)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 48)(32 47)(33 38)(34 37)(35 36)(39 40)(57 69)(58 68)(59 67)(60 66)(61 65)(62 72)(63 71)(64 70)
G:=sub<Sym(72)| (1,25,47)(2,26,48)(3,27,41)(4,28,42)(5,29,43)(6,30,44)(7,31,45)(8,32,46)(9,33,51)(10,34,52)(11,35,53)(12,36,54)(13,37,55)(14,38,56)(15,39,49)(16,40,50)(17,68,57)(18,69,58)(19,70,59)(20,71,60)(21,72,61)(22,65,62)(23,66,63)(24,67,64), (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), (1,69,58)(2,70,59)(3,71,60)(4,72,61)(5,65,62)(6,66,63)(7,67,64)(8,68,57)(9,19,51)(10,20,52)(11,21,53)(12,22,54)(13,23,55)(14,24,56)(15,17,49)(16,18,50)(25,47,40)(26,48,33)(27,41,34)(28,42,35)(29,43,36)(30,44,37)(31,45,38)(32,46,39), (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), (1,8)(2,7)(3,6)(4,5)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,18)(19,24)(20,23)(21,22)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,48)(32,47)(33,38)(34,37)(35,36)(39,40)(57,69)(58,68)(59,67)(60,66)(61,65)(62,72)(63,71)(64,70)>;
G:=Group( (1,25,47)(2,26,48)(3,27,41)(4,28,42)(5,29,43)(6,30,44)(7,31,45)(8,32,46)(9,33,51)(10,34,52)(11,35,53)(12,36,54)(13,37,55)(14,38,56)(15,39,49)(16,40,50)(17,68,57)(18,69,58)(19,70,59)(20,71,60)(21,72,61)(22,65,62)(23,66,63)(24,67,64), (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), (1,69,58)(2,70,59)(3,71,60)(4,72,61)(5,65,62)(6,66,63)(7,67,64)(8,68,57)(9,19,51)(10,20,52)(11,21,53)(12,22,54)(13,23,55)(14,24,56)(15,17,49)(16,18,50)(25,47,40)(26,48,33)(27,41,34)(28,42,35)(29,43,36)(30,44,37)(31,45,38)(32,46,39), (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), (1,8)(2,7)(3,6)(4,5)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,18)(19,24)(20,23)(21,22)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,48)(32,47)(33,38)(34,37)(35,36)(39,40)(57,69)(58,68)(59,67)(60,66)(61,65)(62,72)(63,71)(64,70) );
G=PermutationGroup([[(1,25,47),(2,26,48),(3,27,41),(4,28,42),(5,29,43),(6,30,44),(7,31,45),(8,32,46),(9,33,51),(10,34,52),(11,35,53),(12,36,54),(13,37,55),(14,38,56),(15,39,49),(16,40,50),(17,68,57),(18,69,58),(19,70,59),(20,71,60),(21,72,61),(22,65,62),(23,66,63),(24,67,64)], [(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)], [(1,69,58),(2,70,59),(3,71,60),(4,72,61),(5,65,62),(6,66,63),(7,67,64),(8,68,57),(9,19,51),(10,20,52),(11,21,53),(12,22,54),(13,23,55),(14,24,56),(15,17,49),(16,18,50),(25,47,40),(26,48,33),(27,41,34),(28,42,35),(29,43,36),(30,44,37),(31,45,38),(32,46,39)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,18),(19,24),(20,23),(21,22),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,48),(32,47),(33,38),(34,37),(35,36),(39,40),(57,69),(58,68),(59,67),(60,66),(61,65),(62,72),(63,71),(64,70)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 12A | 12B | 12C | ··· | 12J | 24A | 24B | 24C | 24D | 24E | ··· | 24T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 36 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 6 | 6 | 6 | 6 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
53 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | D12 | D24 | He3⋊C2 | C2×He3⋊C2 | He3⋊5D4 | He3⋊5D8 |
| kernel | He3⋊5D8 | C8×He3 | He3⋊5D4 | C3×C24 | C2×He3 | C3×C12 | He3 | C3×C6 | C32 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 8 | 16 | 4 | 4 | 2 | 4 |
Matrix representation of He3⋊5D8 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 7 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 0 | 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 |
| 0 | 0 | 65 | 64 | 8 |
| 0 | 0 | 65 | 0 | 72 |
| 0 | 0 | 56 | 0 | 8 |
| 55 | 5 | 0 | 0 | 0 |
| 68 | 50 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 18 | 68 | 0 | 0 | 0 |
| 50 | 55 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 7 | 1 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,72,7,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[0,72,0,0,0,1,72,0,0,0,0,0,65,65,56,0,0,64,0,0,0,0,8,72,8],[55,68,0,0,0,5,50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,50,0,0,0,68,55,0,0,0,0,0,1,0,0,0,0,72,72,7,0,0,0,0,1] >;
He3⋊5D8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_5D_8 % in TeX
G:=Group("He3:5D8"); // GroupNames label
G:=SmallGroup(432,176);
// by ID
G=gap.SmallGroup(432,176);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,1124,4037,537]);
// 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,a*d=d*a,e*a*e=a^-1,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