non-abelian, supersoluble, monomial
Aliases: (Q8×He3)⋊6C2, He3⋊5D4⋊6C2, (C3×C12).34D6, (Q8×C32)⋊8S3, He3⋊12(C4○D4), Q8⋊3(He3⋊C2), C32⋊4(Q8⋊3S3), (C2×He3).36C23, (C4×He3).26C22, C3.2(C12.26D6), He3⋊3C4.20C22, C12.51(C2×C3⋊S3), (C4×He3⋊C2)⋊6C2, C6.68(C22×C3⋊S3), C4.7(C2×He3⋊C2), (C3×Q8).18(C3⋊S3), (C3×C6).46(C22×S3), C2.9(C22×He3⋊C2), (C2×He3⋊C2).19C22, SmallGroup(432,395)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊5D4⋊C2
G = < a,b,c,d,e,f | a3=b3=c3=d4=e2=f2=1, ab=ba, cac-1=ab-1, ad=da, eae=faf=a-1, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece=fcf=c-1, ede=d-1, df=fd, fef=d2e >
Subgroups: 845 in 220 conjugacy classes, 53 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, He3, C3×Dic3, C3×C12, S3×C6, Q8⋊3S3, C3×C4○D4, He3⋊C2, C2×He3, S3×C12, C3×D12, Q8×C32, He3⋊3C4, C4×He3, C2×He3⋊C2, C3×Q8⋊3S3, C4×He3⋊C2, He3⋊5D4, Q8×He3, He3⋊5D4⋊C2
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, Q8⋊3S3, He3⋊C2, C22×C3⋊S3, C2×He3⋊C2, C12.26D6, C22×He3⋊C2, He3⋊5D4⋊C2
►On 72 points
Generators in S72
(1 62 29)(2 63 30)(3 64 31)(4 61 32)(5 52 27)(6 49 28)(7 50 25)(8 51 26)(9 33 56)(10 34 53)(11 35 54)(12 36 55)(13 67 20)(14 68 17)(15 65 18)(16 66 19)(21 48 72)(22 45 69)(23 46 70)(24 47 71)(37 42 59)(38 43 60)(39 44 57)(40 41 58)
(1 26 15)(2 27 16)(3 28 13)(4 25 14)(5 66 63)(6 67 64)(7 68 61)(8 65 62)(9 58 69)(10 59 70)(11 60 71)(12 57 72)(17 32 50)(18 29 51)(19 30 52)(20 31 49)(21 36 39)(22 33 40)(23 34 37)(24 35 38)(41 45 56)(42 46 53)(43 47 54)(44 48 55)
(1 65 18)(2 66 19)(3 67 20)(4 68 17)(5 52 16)(6 49 13)(7 50 14)(8 51 15)(9 40 56)(10 37 53)(11 38 54)(12 39 55)(21 44 57)(22 41 58)(23 42 59)(24 43 60)(25 61 32)(26 62 29)(27 63 30)(28 64 31)(33 45 69)(34 46 70)(35 47 71)(36 48 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)
(1 37)(2 40)(3 39)(4 38)(5 69)(6 72)(7 71)(8 70)(9 66)(10 65)(11 68)(12 67)(13 36)(14 35)(15 34)(16 33)(17 54)(18 53)(19 56)(20 55)(21 28)(22 27)(23 26)(24 25)(29 42)(30 41)(31 44)(32 43)(45 52)(46 51)(47 50)(48 49)(57 64)(58 63)(59 62)(60 61)
(5 52)(6 49)(7 50)(8 51)(9 54)(10 55)(11 56)(12 53)(17 68)(18 65)(19 66)(20 67)(21 23)(22 24)(29 62)(30 63)(31 64)(32 61)(33 35)(34 36)(37 39)(38 40)(41 60)(42 57)(43 58)(44 59)(45 71)(46 72)(47 69)(48 70)
G:=sub<Sym(72)| (1,62,29)(2,63,30)(3,64,31)(4,61,32)(5,52,27)(6,49,28)(7,50,25)(8,51,26)(9,33,56)(10,34,53)(11,35,54)(12,36,55)(13,67,20)(14,68,17)(15,65,18)(16,66,19)(21,48,72)(22,45,69)(23,46,70)(24,47,71)(37,42,59)(38,43,60)(39,44,57)(40,41,58), (1,26,15)(2,27,16)(3,28,13)(4,25,14)(5,66,63)(6,67,64)(7,68,61)(8,65,62)(9,58,69)(10,59,70)(11,60,71)(12,57,72)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(21,36,39)(22,33,40)(23,34,37)(24,35,38)(41,45,56)(42,46,53)(43,47,54)(44,48,55), (1,65,18)(2,66,19)(3,67,20)(4,68,17)(5,52,16)(6,49,13)(7,50,14)(8,51,15)(9,40,56)(10,37,53)(11,38,54)(12,39,55)(21,44,57)(22,41,58)(23,42,59)(24,43,60)(25,61,32)(26,62,29)(27,63,30)(28,64,31)(33,45,69)(34,46,70)(35,47,71)(36,48,72), (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), (1,37)(2,40)(3,39)(4,38)(5,69)(6,72)(7,71)(8,70)(9,66)(10,65)(11,68)(12,67)(13,36)(14,35)(15,34)(16,33)(17,54)(18,53)(19,56)(20,55)(21,28)(22,27)(23,26)(24,25)(29,42)(30,41)(31,44)(32,43)(45,52)(46,51)(47,50)(48,49)(57,64)(58,63)(59,62)(60,61), (5,52)(6,49)(7,50)(8,51)(9,54)(10,55)(11,56)(12,53)(17,68)(18,65)(19,66)(20,67)(21,23)(22,24)(29,62)(30,63)(31,64)(32,61)(33,35)(34,36)(37,39)(38,40)(41,60)(42,57)(43,58)(44,59)(45,71)(46,72)(47,69)(48,70)>;
G:=Group( (1,62,29)(2,63,30)(3,64,31)(4,61,32)(5,52,27)(6,49,28)(7,50,25)(8,51,26)(9,33,56)(10,34,53)(11,35,54)(12,36,55)(13,67,20)(14,68,17)(15,65,18)(16,66,19)(21,48,72)(22,45,69)(23,46,70)(24,47,71)(37,42,59)(38,43,60)(39,44,57)(40,41,58), (1,26,15)(2,27,16)(3,28,13)(4,25,14)(5,66,63)(6,67,64)(7,68,61)(8,65,62)(9,58,69)(10,59,70)(11,60,71)(12,57,72)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(21,36,39)(22,33,40)(23,34,37)(24,35,38)(41,45,56)(42,46,53)(43,47,54)(44,48,55), (1,65,18)(2,66,19)(3,67,20)(4,68,17)(5,52,16)(6,49,13)(7,50,14)(8,51,15)(9,40,56)(10,37,53)(11,38,54)(12,39,55)(21,44,57)(22,41,58)(23,42,59)(24,43,60)(25,61,32)(26,62,29)(27,63,30)(28,64,31)(33,45,69)(34,46,70)(35,47,71)(36,48,72), (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), (1,37)(2,40)(3,39)(4,38)(5,69)(6,72)(7,71)(8,70)(9,66)(10,65)(11,68)(12,67)(13,36)(14,35)(15,34)(16,33)(17,54)(18,53)(19,56)(20,55)(21,28)(22,27)(23,26)(24,25)(29,42)(30,41)(31,44)(32,43)(45,52)(46,51)(47,50)(48,49)(57,64)(58,63)(59,62)(60,61), (5,52)(6,49)(7,50)(8,51)(9,54)(10,55)(11,56)(12,53)(17,68)(18,65)(19,66)(20,67)(21,23)(22,24)(29,62)(30,63)(31,64)(32,61)(33,35)(34,36)(37,39)(38,40)(41,60)(42,57)(43,58)(44,59)(45,71)(46,72)(47,69)(48,70) );
G=PermutationGroup([[(1,62,29),(2,63,30),(3,64,31),(4,61,32),(5,52,27),(6,49,28),(7,50,25),(8,51,26),(9,33,56),(10,34,53),(11,35,54),(12,36,55),(13,67,20),(14,68,17),(15,65,18),(16,66,19),(21,48,72),(22,45,69),(23,46,70),(24,47,71),(37,42,59),(38,43,60),(39,44,57),(40,41,58)], [(1,26,15),(2,27,16),(3,28,13),(4,25,14),(5,66,63),(6,67,64),(7,68,61),(8,65,62),(9,58,69),(10,59,70),(11,60,71),(12,57,72),(17,32,50),(18,29,51),(19,30,52),(20,31,49),(21,36,39),(22,33,40),(23,34,37),(24,35,38),(41,45,56),(42,46,53),(43,47,54),(44,48,55)], [(1,65,18),(2,66,19),(3,67,20),(4,68,17),(5,52,16),(6,49,13),(7,50,14),(8,51,15),(9,40,56),(10,37,53),(11,38,54),(12,39,55),(21,44,57),(22,41,58),(23,42,59),(24,43,60),(25,61,32),(26,62,29),(27,63,30),(28,64,31),(33,45,69),(34,46,70),(35,47,71),(36,48,72)], [(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)], [(1,37),(2,40),(3,39),(4,38),(5,69),(6,72),(7,71),(8,70),(9,66),(10,65),(11,68),(12,67),(13,36),(14,35),(15,34),(16,33),(17,54),(18,53),(19,56),(20,55),(21,28),(22,27),(23,26),(24,25),(29,42),(30,41),(31,44),(32,43),(45,52),(46,51),(47,50),(48,49),(57,64),(58,63),(59,62),(60,61)], [(5,52),(6,49),(7,50),(8,51),(9,54),(10,55),(11,56),(12,53),(17,68),(18,65),(19,66),(20,67),(21,23),(22,24),(29,62),(30,63),(31,64),(32,61),(33,35),(34,36),(37,39),(38,40),(41,60),(42,57),(43,58),(44,59),(45,71),(46,72),(47,69),(48,70)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6L | 12A | ··· | 12F | 12G | 12H | 12I | 12J | 12K | ··· | 12V |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 18 | 18 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 9 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 18 | ··· | 18 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 12 | ··· | 12 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 |
| type | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D6 | C4○D4 | He3⋊C2 | C2×He3⋊C2 | Q8⋊3S3 | He3⋊5D4⋊C2 |
| kernel | He3⋊5D4⋊C2 | C4×He3⋊C2 | He3⋊5D4 | Q8×He3 | Q8×C32 | C3×C12 | He3 | Q8 | C4 | C32 | C1 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 2 | 4 | 12 | 4 | 4 |
Matrix representation of He3⋊5D4⋊C2 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 8 | 10 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,4,0,0,3,0,0],[8,8,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,8,0,0,0,10,5,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12],[1,1,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12] >;
He3⋊5D4⋊C2 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_5D_4\rtimes C_2 % in TeX
G:=Group("He3:5D4:C2"); // GroupNames label
G:=SmallGroup(432,395);
// by ID
G=gap.SmallGroup(432,395);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^4=e^2=f^2=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=f*a*f=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,e*c*e=f*c*f=c^-1,e*d*e=d^-1,d*f=f*d,f*e*f=d^2*e>;
// generators/relations