metabelian, supersoluble, monomial
Aliases: He3⋊6SD16, C24⋊2S3⋊C3, (C3×C24)⋊3S3, (C3×C24)⋊2C6, C24.9(C3×S3), (C8×He3)⋊3C2, C6.5(C3×D12), C12.68(S3×C6), C8⋊2(C32⋊C6), (C3×C12).41D6, (C3×C6).14D12, C32⋊4Q8⋊1C6, C12⋊S3.1C6, He3⋊3Q8⋊10C2, (C2×He3).18D4, He3⋊4D4.4C2, C32⋊3(C3×SD16), C32⋊5(C24⋊C2), C2.4(He3⋊4D4), (C4×He3).33C22, (C3×C6).7(C3×D4), C3.2(C3×C24⋊C2), (C3×C12).9(C2×C6), C4.9(C2×C32⋊C6), SmallGroup(432,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊6SD16
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=d3 >
Subgroups: 461 in 78 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C24, C24, Dic6, D12, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, C3×SD16, C32⋊C6, C2×He3, C3×C24, C3×C24, C3×Dic6, C3×D12, C32⋊4Q8, C12⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C3×C24⋊C2, C24⋊2S3, C8×He3, He3⋊3Q8, He3⋊4D4, He3⋊6SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, D12, C3×D4, S3×C6, C24⋊C2, C3×SD16, C32⋊C6, C3×D12, C2×C32⋊C6, C3×C24⋊C2, He3⋊4D4, He3⋊6SD16
►On 72 points
Generators in S72
(9 26 47)(10 27 48)(11 28 41)(12 29 42)(13 30 43)(14 31 44)(15 32 45)(16 25 46)(17 55 36)(18 56 37)(19 49 38)(20 50 39)(21 51 40)(22 52 33)(23 53 34)(24 54 35)
(1 71 58)(2 72 59)(3 65 60)(4 66 61)(5 67 62)(6 68 63)(7 69 64)(8 70 57)(9 47 26)(10 48 27)(11 41 28)(12 42 29)(13 43 30)(14 44 31)(15 45 32)(16 46 25)(17 55 36)(18 56 37)(19 49 38)(20 50 39)(21 51 40)(22 52 33)(23 53 34)(24 54 35)
(1 15 24)(2 16 17)(3 9 18)(4 10 19)(5 11 20)(6 12 21)(7 13 22)(8 14 23)(25 36 59)(26 37 60)(27 38 61)(28 39 62)(29 40 63)(30 33 64)(31 34 57)(32 35 58)(41 50 67)(42 51 68)(43 52 69)(44 53 70)(45 54 71)(46 55 72)(47 56 65)(48 49 66)
(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 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 19)(18 22)(21 23)(25 48)(26 43)(27 46)(28 41)(29 44)(30 47)(31 42)(32 45)(33 56)(34 51)(35 54)(36 49)(37 52)(38 55)(39 50)(40 53)(57 68)(58 71)(59 66)(60 69)(61 72)(62 67)(63 70)(64 65)
G:=sub<Sym(72)| (9,26,47)(10,27,48)(11,28,41)(12,29,42)(13,30,43)(14,31,44)(15,32,45)(16,25,46)(17,55,36)(18,56,37)(19,49,38)(20,50,39)(21,51,40)(22,52,33)(23,53,34)(24,54,35), (1,71,58)(2,72,59)(3,65,60)(4,66,61)(5,67,62)(6,68,63)(7,69,64)(8,70,57)(9,47,26)(10,48,27)(11,41,28)(12,42,29)(13,43,30)(14,44,31)(15,45,32)(16,46,25)(17,55,36)(18,56,37)(19,49,38)(20,50,39)(21,51,40)(22,52,33)(23,53,34)(24,54,35), (1,15,24)(2,16,17)(3,9,18)(4,10,19)(5,11,20)(6,12,21)(7,13,22)(8,14,23)(25,36,59)(26,37,60)(27,38,61)(28,39,62)(29,40,63)(30,33,64)(31,34,57)(32,35,58)(41,50,67)(42,51,68)(43,52,69)(44,53,70)(45,54,71)(46,55,72)(47,56,65)(48,49,66), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,19)(18,22)(21,23)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45)(33,56)(34,51)(35,54)(36,49)(37,52)(38,55)(39,50)(40,53)(57,68)(58,71)(59,66)(60,69)(61,72)(62,67)(63,70)(64,65)>;
G:=Group( (9,26,47)(10,27,48)(11,28,41)(12,29,42)(13,30,43)(14,31,44)(15,32,45)(16,25,46)(17,55,36)(18,56,37)(19,49,38)(20,50,39)(21,51,40)(22,52,33)(23,53,34)(24,54,35), (1,71,58)(2,72,59)(3,65,60)(4,66,61)(5,67,62)(6,68,63)(7,69,64)(8,70,57)(9,47,26)(10,48,27)(11,41,28)(12,42,29)(13,43,30)(14,44,31)(15,45,32)(16,46,25)(17,55,36)(18,56,37)(19,49,38)(20,50,39)(21,51,40)(22,52,33)(23,53,34)(24,54,35), (1,15,24)(2,16,17)(3,9,18)(4,10,19)(5,11,20)(6,12,21)(7,13,22)(8,14,23)(25,36,59)(26,37,60)(27,38,61)(28,39,62)(29,40,63)(30,33,64)(31,34,57)(32,35,58)(41,50,67)(42,51,68)(43,52,69)(44,53,70)(45,54,71)(46,55,72)(47,56,65)(48,49,66), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,19)(18,22)(21,23)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45)(33,56)(34,51)(35,54)(36,49)(37,52)(38,55)(39,50)(40,53)(57,68)(58,71)(59,66)(60,69)(61,72)(62,67)(63,70)(64,65) );
G=PermutationGroup([[(9,26,47),(10,27,48),(11,28,41),(12,29,42),(13,30,43),(14,31,44),(15,32,45),(16,25,46),(17,55,36),(18,56,37),(19,49,38),(20,50,39),(21,51,40),(22,52,33),(23,53,34),(24,54,35)], [(1,71,58),(2,72,59),(3,65,60),(4,66,61),(5,67,62),(6,68,63),(7,69,64),(8,70,57),(9,47,26),(10,48,27),(11,41,28),(12,42,29),(13,43,30),(14,44,31),(15,45,32),(16,46,25),(17,55,36),(18,56,37),(19,49,38),(20,50,39),(21,51,40),(22,52,33),(23,53,34),(24,54,35)], [(1,15,24),(2,16,17),(3,9,18),(4,10,19),(5,11,20),(6,12,21),(7,13,22),(8,14,23),(25,36,59),(26,37,60),(27,38,61),(28,39,62),(29,40,63),(30,33,64),(31,34,57),(32,35,58),(41,50,67),(42,51,68),(43,52,69),(44,53,70),(45,54,71),(46,55,72),(47,56,65),(48,49,66)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,19),(18,22),(21,23),(25,48),(26,43),(27,46),(28,41),(29,44),(30,47),(31,42),(32,45),(33,56),(34,51),(35,54),(36,49),(37,52),(38,55),(39,50),(40,53),(57,68),(58,71),(59,66),(60,69),(61,72),(62,67),(63,70),(64,65)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | 12B | 12C | ··· | 12J | 12K | 12L | 24A | 24B | 24C | 24D | 24E | ··· | 24T |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 36 | 2 | 3 | 3 | 6 | 6 | 6 | 36 | 36 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 36 | 36 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
53 irreducible representations
| dim | 1 | 1 | 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 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | SD16 | C3×S3 | D12 | C3×D4 | S3×C6 | C24⋊C2 | C3×SD16 | C3×D12 | C3×C24⋊C2 | C32⋊C6 | C2×C32⋊C6 | He3⋊4D4 | He3⋊6SD16 |
| kernel | He3⋊6SD16 | C8×He3 | He3⋊3Q8 | He3⋊4D4 | C24⋊2S3 | C3×C24 | C32⋊4Q8 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of He3⋊6SD16 ►in GL8(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 64 | 72 | 72 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 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 | 9 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 64 | 72 | 72 | 0 | 0 |
| 0 | 0 | 65 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 8 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 64 | 64 | 71 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 65 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 65 | 0 | 0 | 0 |
| 48 | 62 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 37 | 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 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 64 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 | 72 | 72 |
G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,64,8,0,0,0,0,1,0,64,8,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,9,0,65,0,0,0,1,0,0,64,0,8,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,64,0,0,0,1,0,0,0,64,0,0,0,0,0,0,72,71,9,9,65,65,0,0,1,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[48,11,0,0,0,0,0,0,62,37,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,64,0,8,0,0,1,0,0,64,0,8,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72] >;
He3⋊6SD16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_6{\rm SD}_{16} % in TeX
G:=Group("He3:6SD16"); // GroupNames label
G:=SmallGroup(432,117);
// by ID
G=gap.SmallGroup(432,117);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,4037,2035,14118]);
// 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^3>;
// generators/relations