metabelian, supersoluble, monomial
Aliases: He3⋊4D8, C32⋊3D24, C32⋊5D8⋊C3, (C3×C24)⋊1C6, (C3×C24)⋊1S3, C24.4(C3×S3), (C8×He3)⋊1C2, C6.6(C3×D12), C3.2(C3×D24), C12⋊S3⋊1C6, C32⋊2(C3×D8), C12.69(S3×C6), C8⋊1(C32⋊C6), (C3×C12).42D6, (C3×C6).15D12, He3⋊4D4⋊10C2, (C2×He3).19D4, C2.5(He3⋊4D4), (C4×He3).34C22, (C3×C6).8(C3×D4), (C3×C12).10(C2×C6), C4.10(C2×C32⋊C6), SmallGroup(432,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊4D8
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, ebe=b-1, cd=dc, ce=ec, ede=d-1 >
Subgroups: 601 in 85 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C24, C24, D12, C3×D4, He3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, D24, C3×D8, C32⋊C6, C2×He3, C3×C24, C3×C24, C3×D12, C12⋊S3, C4×He3, C2×C32⋊C6, C3×D24, C32⋊5D8, C8×He3, He3⋊4D4, He3⋊4D8
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, D12, C3×D4, S3×C6, D24, C3×D8, C32⋊C6, C3×D12, C2×C32⋊C6, C3×D24, He3⋊4D4, He3⋊4D8
►On 72 points
Generators in S72
(1 61 22)(2 62 23)(3 63 24)(4 64 17)(5 57 18)(6 58 19)(7 59 20)(8 60 21)(9 52 37)(10 53 38)(11 54 39)(12 55 40)(13 56 33)(14 49 34)(15 50 35)(16 51 36)(25 48 70)(26 41 71)(27 42 72)(28 43 65)(29 44 66)(30 45 67)(31 46 68)(32 47 69)
(1 15 45)(2 16 46)(3 9 47)(4 10 48)(5 11 41)(6 12 42)(7 13 43)(8 14 44)(17 38 25)(18 39 26)(19 40 27)(20 33 28)(21 34 29)(22 35 30)(23 36 31)(24 37 32)(49 66 60)(50 67 61)(51 68 62)(52 69 63)(53 70 64)(54 71 57)(55 72 58)(56 65 59)
(17 25 38)(18 26 39)(19 27 40)(20 28 33)(21 29 34)(22 30 35)(23 31 36)(24 32 37)(49 66 60)(50 67 61)(51 68 62)(52 69 63)(53 70 64)(54 71 57)(55 72 58)(56 65 59)
(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 7)(2 6)(3 5)(9 41)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 53)(26 52)(27 51)(28 50)(29 49)(30 56)(31 55)(32 54)(33 67)(34 66)(35 65)(36 72)(37 71)(38 70)(39 69)(40 68)
G:=sub<Sym(72)| (1,61,22)(2,62,23)(3,63,24)(4,64,17)(5,57,18)(6,58,19)(7,59,20)(8,60,21)(9,52,37)(10,53,38)(11,54,39)(12,55,40)(13,56,33)(14,49,34)(15,50,35)(16,51,36)(25,48,70)(26,41,71)(27,42,72)(28,43,65)(29,44,66)(30,45,67)(31,46,68)(32,47,69), (1,15,45)(2,16,46)(3,9,47)(4,10,48)(5,11,41)(6,12,42)(7,13,43)(8,14,44)(17,38,25)(18,39,26)(19,40,27)(20,33,28)(21,34,29)(22,35,30)(23,36,31)(24,37,32)(49,66,60)(50,67,61)(51,68,62)(52,69,63)(53,70,64)(54,71,57)(55,72,58)(56,65,59), (17,25,38)(18,26,39)(19,27,40)(20,28,33)(21,29,34)(22,30,35)(23,31,36)(24,32,37)(49,66,60)(50,67,61)(51,68,62)(52,69,63)(53,70,64)(54,71,57)(55,72,58)(56,65,59), (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), (1,7)(2,6)(3,5)(9,41)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,53)(26,52)(27,51)(28,50)(29,49)(30,56)(31,55)(32,54)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68)>;
G:=Group( (1,61,22)(2,62,23)(3,63,24)(4,64,17)(5,57,18)(6,58,19)(7,59,20)(8,60,21)(9,52,37)(10,53,38)(11,54,39)(12,55,40)(13,56,33)(14,49,34)(15,50,35)(16,51,36)(25,48,70)(26,41,71)(27,42,72)(28,43,65)(29,44,66)(30,45,67)(31,46,68)(32,47,69), (1,15,45)(2,16,46)(3,9,47)(4,10,48)(5,11,41)(6,12,42)(7,13,43)(8,14,44)(17,38,25)(18,39,26)(19,40,27)(20,33,28)(21,34,29)(22,35,30)(23,36,31)(24,37,32)(49,66,60)(50,67,61)(51,68,62)(52,69,63)(53,70,64)(54,71,57)(55,72,58)(56,65,59), (17,25,38)(18,26,39)(19,27,40)(20,28,33)(21,29,34)(22,30,35)(23,31,36)(24,32,37)(49,66,60)(50,67,61)(51,68,62)(52,69,63)(53,70,64)(54,71,57)(55,72,58)(56,65,59), (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), (1,7)(2,6)(3,5)(9,41)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,53)(26,52)(27,51)(28,50)(29,49)(30,56)(31,55)(32,54)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68) );
G=PermutationGroup([[(1,61,22),(2,62,23),(3,63,24),(4,64,17),(5,57,18),(6,58,19),(7,59,20),(8,60,21),(9,52,37),(10,53,38),(11,54,39),(12,55,40),(13,56,33),(14,49,34),(15,50,35),(16,51,36),(25,48,70),(26,41,71),(27,42,72),(28,43,65),(29,44,66),(30,45,67),(31,46,68),(32,47,69)], [(1,15,45),(2,16,46),(3,9,47),(4,10,48),(5,11,41),(6,12,42),(7,13,43),(8,14,44),(17,38,25),(18,39,26),(19,40,27),(20,33,28),(21,34,29),(22,35,30),(23,36,31),(24,37,32),(49,66,60),(50,67,61),(51,68,62),(52,69,63),(53,70,64),(54,71,57),(55,72,58),(56,65,59)], [(17,25,38),(18,26,39),(19,27,40),(20,28,33),(21,29,34),(22,30,35),(23,31,36),(24,32,37),(49,66,60),(50,67,61),(51,68,62),(52,69,63),(53,70,64),(54,71,57),(55,72,58),(56,65,59)], [(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)], [(1,7),(2,6),(3,5),(9,41),(10,48),(11,47),(12,46),(13,45),(14,44),(15,43),(16,42),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,53),(26,52),(27,51),(28,50),(29,49),(30,56),(31,55),(32,54),(33,67),(34,66),(35,65),(36,72),(37,71),(38,70),(39,69),(40,68)]])
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 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 2 | 3 | 3 | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | D8 | C3×S3 | D12 | C3×D4 | S3×C6 | D24 | C3×D8 | C3×D12 | C3×D24 | C32⋊C6 | C2×C32⋊C6 | He3⋊4D4 | He3⋊4D8 |
| kernel | He3⋊4D8 | C8×He3 | He3⋊4D4 | C32⋊5D8 | C3×C24 | C12⋊S3 | C3×C24 | C2×He3 | C3×C12 | He3 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of He3⋊4D8 ►in GL8(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 |
| 50 | 68 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 55 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 66 | 59 | 0 | 0 | 0 | 0 | 0 | 0 |
| 66 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
G:=sub<GL(8,GF(73))| [0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[50,5,0,0,0,0,0,0,68,55,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[66,66,0,0,0,0,0,0,59,7,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0] >;
He3⋊4D8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_4D_8 % in TeX
G:=Group("He3:4D8"); // GroupNames label
G:=SmallGroup(432,118);
// by ID
G=gap.SmallGroup(432,118);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,260,1011,80,4037,2035,14118]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations