non-abelian, supersoluble, monomial
Aliases: He3⋊3(C2×Q8), C12.85(S32), C32⋊2(S3×Q8), He3⋊2Q8⋊3C2, He3⋊3Q8⋊6C2, He3⋊C2⋊2Q8, (C3×C12).23D6, C3⋊Dic3.2D6, C32⋊4Q8⋊5S3, C4.12(C32⋊D6), (C2×He3).5C23, C32⋊C12.2C22, (C4×He3).19C22, C3.2(Dic3.D6), He3⋊3C4.12C22, C6.79(C2×S32), C2.8(C2×C32⋊D6), (C3×C6).5(C22×S3), He3⋊(C2×C4).1C2, (C4×He3⋊C2).2C2, (C2×He3⋊C2).13C22, SmallGroup(432,298)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊3(C2×Q8)
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, cac-1=ab-1, dad=eae-1=faf-1=a-1, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd=ece-1=c-1, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 787 in 149 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C4 [×5], C22, S3 [×6], C6, C6 [×5], C2×C4 [×3], Q8 [×4], C32 [×2], C32, Dic3 [×11], C12, C12 [×8], D6 [×3], C2×C6, C2×Q8, C3×S3 [×6], C3×C6 [×2], C3×C6, Dic6 [×10], C4×S3 [×7], C2×Dic3 [×2], C2×C12, C3×Q8 [×2], He3, C3×Dic3 [×7], C3⋊Dic3 [×4], C3×C12 [×2], C3×C12, S3×C6 [×3], C2×Dic6, S3×Q8 [×2], He3⋊C2 [×2], C2×He3, S3×Dic3 [×4], C32⋊2Q8 [×4], C3×Dic6 [×2], S3×C12 [×3], C32⋊4Q8 [×2], C32⋊C12 [×4], He3⋊3C4, C4×He3, C2×He3⋊C2, S3×Dic6 [×2], He3⋊2Q8 [×2], He3⋊(C2×C4) [×2], He3⋊3Q8 [×2], C4×He3⋊C2, He3⋊3(C2×Q8)
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], Q8 [×2], C23, D6 [×6], C2×Q8, C22×S3 [×2], S32, S3×Q8 [×2], C2×S32, C32⋊D6, Dic3.D6, C2×C32⋊D6, He3⋊3(C2×Q8)
►On 72 points
Generators in S72
(1 8 21)(2 22 5)(3 6 23)(4 24 7)(9 51 68)(10 65 52)(11 49 66)(12 67 50)(13 46 37)(14 38 47)(15 48 39)(16 40 45)(17 42 56)(18 53 43)(19 44 54)(20 55 41)(25 61 71)(26 72 62)(27 63 69)(28 70 64)(29 57 36)(30 33 58)(31 59 34)(32 35 60)
(1 30 16)(2 31 13)(3 32 14)(4 29 15)(5 34 37)(6 35 38)(7 36 39)(8 33 40)(9 43 63)(10 44 64)(11 41 61)(12 42 62)(17 72 50)(18 69 51)(19 70 52)(20 71 49)(21 58 45)(22 59 46)(23 60 47)(24 57 48)(25 66 55)(26 67 56)(27 68 53)(28 65 54)
(5 37 34)(6 35 38)(7 39 36)(8 33 40)(17 50 72)(18 69 51)(19 52 70)(20 71 49)(21 45 58)(22 59 46)(23 47 60)(24 57 48)(25 55 66)(26 67 56)(27 53 68)(28 65 54)
(1 3)(2 4)(5 24)(6 21)(7 22)(8 23)(9 11)(10 12)(13 15)(14 16)(17 54)(18 55)(19 56)(20 53)(25 69)(26 70)(27 71)(28 72)(29 31)(30 32)(33 60)(34 57)(35 58)(36 59)(37 48)(38 45)(39 46)(40 47)(41 43)(42 44)(49 68)(50 65)(51 66)(52 67)(61 63)(62 64)
(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 9 3 11)(2 12 4 10)(5 67 7 65)(6 66 8 68)(13 42 15 44)(14 41 16 43)(17 48 19 46)(18 47 20 45)(21 51 23 49)(22 50 24 52)(25 33 27 35)(26 36 28 34)(29 64 31 62)(30 63 32 61)(37 56 39 54)(38 55 40 53)(57 70 59 72)(58 69 60 71)
G:=sub<Sym(72)| (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,51,68)(10,65,52)(11,49,66)(12,67,50)(13,46,37)(14,38,47)(15,48,39)(16,40,45)(17,42,56)(18,53,43)(19,44,54)(20,55,41)(25,61,71)(26,72,62)(27,63,69)(28,70,64)(29,57,36)(30,33,58)(31,59,34)(32,35,60), (1,30,16)(2,31,13)(3,32,14)(4,29,15)(5,34,37)(6,35,38)(7,36,39)(8,33,40)(9,43,63)(10,44,64)(11,41,61)(12,42,62)(17,72,50)(18,69,51)(19,70,52)(20,71,49)(21,58,45)(22,59,46)(23,60,47)(24,57,48)(25,66,55)(26,67,56)(27,68,53)(28,65,54), (5,37,34)(6,35,38)(7,39,36)(8,33,40)(17,50,72)(18,69,51)(19,52,70)(20,71,49)(21,45,58)(22,59,46)(23,47,60)(24,57,48)(25,55,66)(26,67,56)(27,53,68)(28,65,54), (1,3)(2,4)(5,24)(6,21)(7,22)(8,23)(9,11)(10,12)(13,15)(14,16)(17,54)(18,55)(19,56)(20,53)(25,69)(26,70)(27,71)(28,72)(29,31)(30,32)(33,60)(34,57)(35,58)(36,59)(37,48)(38,45)(39,46)(40,47)(41,43)(42,44)(49,68)(50,65)(51,66)(52,67)(61,63)(62,64), (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), (1,9,3,11)(2,12,4,10)(5,67,7,65)(6,66,8,68)(13,42,15,44)(14,41,16,43)(17,48,19,46)(18,47,20,45)(21,51,23,49)(22,50,24,52)(25,33,27,35)(26,36,28,34)(29,64,31,62)(30,63,32,61)(37,56,39,54)(38,55,40,53)(57,70,59,72)(58,69,60,71)>;
G:=Group( (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,51,68)(10,65,52)(11,49,66)(12,67,50)(13,46,37)(14,38,47)(15,48,39)(16,40,45)(17,42,56)(18,53,43)(19,44,54)(20,55,41)(25,61,71)(26,72,62)(27,63,69)(28,70,64)(29,57,36)(30,33,58)(31,59,34)(32,35,60), (1,30,16)(2,31,13)(3,32,14)(4,29,15)(5,34,37)(6,35,38)(7,36,39)(8,33,40)(9,43,63)(10,44,64)(11,41,61)(12,42,62)(17,72,50)(18,69,51)(19,70,52)(20,71,49)(21,58,45)(22,59,46)(23,60,47)(24,57,48)(25,66,55)(26,67,56)(27,68,53)(28,65,54), (5,37,34)(6,35,38)(7,39,36)(8,33,40)(17,50,72)(18,69,51)(19,52,70)(20,71,49)(21,45,58)(22,59,46)(23,47,60)(24,57,48)(25,55,66)(26,67,56)(27,53,68)(28,65,54), (1,3)(2,4)(5,24)(6,21)(7,22)(8,23)(9,11)(10,12)(13,15)(14,16)(17,54)(18,55)(19,56)(20,53)(25,69)(26,70)(27,71)(28,72)(29,31)(30,32)(33,60)(34,57)(35,58)(36,59)(37,48)(38,45)(39,46)(40,47)(41,43)(42,44)(49,68)(50,65)(51,66)(52,67)(61,63)(62,64), (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), (1,9,3,11)(2,12,4,10)(5,67,7,65)(6,66,8,68)(13,42,15,44)(14,41,16,43)(17,48,19,46)(18,47,20,45)(21,51,23,49)(22,50,24,52)(25,33,27,35)(26,36,28,34)(29,64,31,62)(30,63,32,61)(37,56,39,54)(38,55,40,53)(57,70,59,72)(58,69,60,71) );
G=PermutationGroup([(1,8,21),(2,22,5),(3,6,23),(4,24,7),(9,51,68),(10,65,52),(11,49,66),(12,67,50),(13,46,37),(14,38,47),(15,48,39),(16,40,45),(17,42,56),(18,53,43),(19,44,54),(20,55,41),(25,61,71),(26,72,62),(27,63,69),(28,70,64),(29,57,36),(30,33,58),(31,59,34),(32,35,60)], [(1,30,16),(2,31,13),(3,32,14),(4,29,15),(5,34,37),(6,35,38),(7,36,39),(8,33,40),(9,43,63),(10,44,64),(11,41,61),(12,42,62),(17,72,50),(18,69,51),(19,70,52),(20,71,49),(21,58,45),(22,59,46),(23,60,47),(24,57,48),(25,66,55),(26,67,56),(27,68,53),(28,65,54)], [(5,37,34),(6,35,38),(7,39,36),(8,33,40),(17,50,72),(18,69,51),(19,52,70),(20,71,49),(21,45,58),(22,59,46),(23,47,60),(24,57,48),(25,55,66),(26,67,56),(27,53,68),(28,65,54)], [(1,3),(2,4),(5,24),(6,21),(7,22),(8,23),(9,11),(10,12),(13,15),(14,16),(17,54),(18,55),(19,56),(20,53),(25,69),(26,70),(27,71),(28,72),(29,31),(30,32),(33,60),(34,57),(35,58),(36,59),(37,48),(38,45),(39,46),(40,47),(41,43),(42,44),(49,68),(50,65),(51,66),(52,67),(61,63),(62,64)], [(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)], [(1,9,3,11),(2,12,4,10),(5,67,7,65),(6,66,8,68),(13,42,15,44),(14,41,16,43),(17,48,19,46),(18,47,20,45),(21,51,23,49),(22,50,24,52),(25,33,27,35),(26,36,28,34),(29,64,31,62),(30,63,32,61),(37,56,39,54),(38,55,40,53),(57,70,59,72),(58,69,60,71)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | ··· | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 9 | 9 | 2 | 6 | 6 | 12 | 2 | 18 | ··· | 18 | 2 | 6 | 6 | 12 | 18 | 18 | 2 | 2 | 12 | 12 | 12 | 12 | 18 | 18 | 36 | 36 | 36 | 36 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | S32 | S3×Q8 | C2×S32 | Dic3.D6 | C32⋊D6 | C2×C32⋊D6 | He3⋊3(C2×Q8) |
| kernel | He3⋊3(C2×Q8) | He3⋊2Q8 | He3⋊(C2×C4) | He3⋊3Q8 | C4×He3⋊C2 | C32⋊4Q8 | He3⋊C2 | C3⋊Dic3 | C3×C12 | C12 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of He3⋊3(C2×Q8) ►in GL6(𝔽13)
| 0 | 0 | 12 | 1 | 0 | 0 |
| 1 | 1 | 11 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 1 | 0 | 12 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 12 | 12 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 10 | 7 |
| 0 | 7 | 0 | 0 | 6 | 3 |
| 6 | 0 | 10 | 7 | 0 | 0 |
| 0 | 7 | 6 | 3 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 4 | 2 |
| 2 | 4 | 0 | 0 | 11 | 9 |
| 11 | 0 | 4 | 2 | 0 | 0 |
| 2 | 4 | 11 | 9 | 0 | 0 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,1,0,1,0,0,0,0,12,11,12,12,12,12,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,12,0,12,0,1,0,0,1,0,1,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,1,0,0,1,0,1,1,0,0,1,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,12,0,0],[3,6,6,0,6,0,7,10,0,7,0,7,0,0,0,0,10,6,0,0,0,0,7,3,0,0,10,6,0,0,0,0,7,3,0,0],[11,2,11,2,11,2,4,2,0,4,0,4,0,0,0,0,4,11,0,0,0,0,2,9,0,0,4,11,0,0,0,0,2,9,0,0] >;
He3⋊3(C2×Q8) in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_3(C_2\times Q_8) % in TeX
G:=Group("He3:3(C2xQ8)"); // GroupNames label
G:=SmallGroup(432,298);
// by ID
G=gap.SmallGroup(432,298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,135,58,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^4=1,f^2=e^2,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=e*a*e^-1=f*a*f^-1=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b^-1,d*c*d=e*c*e^-1=c^-1,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations