non-abelian, supersoluble, monomial
Aliases: C12.85S32, He3⋊3(C2×Q8), 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), He3⋊(C2×C4).1C2, (C3×C6).5(C22×S3), (C4×He3⋊C2).2C2, (C2×He3⋊C2).13C22, SmallGroup(432,298)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.85S32
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, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×Dic6, S3×Q8, He3⋊C2, C2×He3, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C32⋊C12, He3⋊3C4, C4×He3, C2×He3⋊C2, S3×Dic6, He3⋊2Q8, He3⋊(C2×C4), He3⋊3Q8, C4×He3⋊C2, C12.85S32
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C22×S3, S32, S3×Q8, C2×S32, C32⋊D6, Dic3.D6, C2×C32⋊D6, C12.85S32
►On 72 points
Generators in S72
(1 8 45)(2 46 5)(3 6 47)(4 48 7)(9 66 43)(10 44 67)(11 68 41)(12 42 65)(13 21 37)(14 38 22)(15 23 39)(16 40 24)(17 60 62)(18 63 57)(19 58 64)(20 61 59)(25 49 71)(26 72 50)(27 51 69)(28 70 52)(29 55 36)(30 33 56)(31 53 34)(32 35 54)
(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 69 63)(10 70 64)(11 71 61)(12 72 62)(17 42 50)(18 43 51)(19 44 52)(20 41 49)(21 46 53)(22 47 54)(23 48 55)(24 45 56)(25 59 68)(26 60 65)(27 57 66)(28 58 67)
(5 37 34)(6 35 38)(7 39 36)(8 33 40)(9 69 63)(10 64 70)(11 71 61)(12 62 72)(21 46 53)(22 54 47)(23 48 55)(24 56 45)(25 68 59)(26 60 65)(27 66 57)(28 58 67)
(1 3)(2 4)(5 48)(6 45)(7 46)(8 47)(9 68)(10 65)(11 66)(12 67)(13 15)(14 16)(17 19)(18 20)(21 39)(22 40)(23 37)(24 38)(25 69)(26 70)(27 71)(28 72)(29 31)(30 32)(33 54)(34 55)(35 56)(36 53)(41 43)(42 44)(49 51)(50 52)(57 61)(58 62)(59 63)(60 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 18 3 20)(2 17 4 19)(5 60 7 58)(6 59 8 57)(9 22 11 24)(10 21 12 23)(13 42 15 44)(14 41 16 43)(25 33 27 35)(26 36 28 34)(29 52 31 50)(30 51 32 49)(37 65 39 67)(38 68 40 66)(45 63 47 61)(46 62 48 64)(53 72 55 70)(54 71 56 69)
G:=sub<Sym(72)| (1,8,45)(2,46,5)(3,6,47)(4,48,7)(9,66,43)(10,44,67)(11,68,41)(12,42,65)(13,21,37)(14,38,22)(15,23,39)(16,40,24)(17,60,62)(18,63,57)(19,58,64)(20,61,59)(25,49,71)(26,72,50)(27,51,69)(28,70,52)(29,55,36)(30,33,56)(31,53,34)(32,35,54), (1,30,16)(2,31,13)(3,32,14)(4,29,15)(5,34,37)(6,35,38)(7,36,39)(8,33,40)(9,69,63)(10,70,64)(11,71,61)(12,72,62)(17,42,50)(18,43,51)(19,44,52)(20,41,49)(21,46,53)(22,47,54)(23,48,55)(24,45,56)(25,59,68)(26,60,65)(27,57,66)(28,58,67), (5,37,34)(6,35,38)(7,39,36)(8,33,40)(9,69,63)(10,64,70)(11,71,61)(12,62,72)(21,46,53)(22,54,47)(23,48,55)(24,56,45)(25,68,59)(26,60,65)(27,66,57)(28,58,67), (1,3)(2,4)(5,48)(6,45)(7,46)(8,47)(9,68)(10,65)(11,66)(12,67)(13,15)(14,16)(17,19)(18,20)(21,39)(22,40)(23,37)(24,38)(25,69)(26,70)(27,71)(28,72)(29,31)(30,32)(33,54)(34,55)(35,56)(36,53)(41,43)(42,44)(49,51)(50,52)(57,61)(58,62)(59,63)(60,64), (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), (1,18,3,20)(2,17,4,19)(5,60,7,58)(6,59,8,57)(9,22,11,24)(10,21,12,23)(13,42,15,44)(14,41,16,43)(25,33,27,35)(26,36,28,34)(29,52,31,50)(30,51,32,49)(37,65,39,67)(38,68,40,66)(45,63,47,61)(46,62,48,64)(53,72,55,70)(54,71,56,69)>;
G:=Group( (1,8,45)(2,46,5)(3,6,47)(4,48,7)(9,66,43)(10,44,67)(11,68,41)(12,42,65)(13,21,37)(14,38,22)(15,23,39)(16,40,24)(17,60,62)(18,63,57)(19,58,64)(20,61,59)(25,49,71)(26,72,50)(27,51,69)(28,70,52)(29,55,36)(30,33,56)(31,53,34)(32,35,54), (1,30,16)(2,31,13)(3,32,14)(4,29,15)(5,34,37)(6,35,38)(7,36,39)(8,33,40)(9,69,63)(10,70,64)(11,71,61)(12,72,62)(17,42,50)(18,43,51)(19,44,52)(20,41,49)(21,46,53)(22,47,54)(23,48,55)(24,45,56)(25,59,68)(26,60,65)(27,57,66)(28,58,67), (5,37,34)(6,35,38)(7,39,36)(8,33,40)(9,69,63)(10,64,70)(11,71,61)(12,62,72)(21,46,53)(22,54,47)(23,48,55)(24,56,45)(25,68,59)(26,60,65)(27,66,57)(28,58,67), (1,3)(2,4)(5,48)(6,45)(7,46)(8,47)(9,68)(10,65)(11,66)(12,67)(13,15)(14,16)(17,19)(18,20)(21,39)(22,40)(23,37)(24,38)(25,69)(26,70)(27,71)(28,72)(29,31)(30,32)(33,54)(34,55)(35,56)(36,53)(41,43)(42,44)(49,51)(50,52)(57,61)(58,62)(59,63)(60,64), (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), (1,18,3,20)(2,17,4,19)(5,60,7,58)(6,59,8,57)(9,22,11,24)(10,21,12,23)(13,42,15,44)(14,41,16,43)(25,33,27,35)(26,36,28,34)(29,52,31,50)(30,51,32,49)(37,65,39,67)(38,68,40,66)(45,63,47,61)(46,62,48,64)(53,72,55,70)(54,71,56,69) );
G=PermutationGroup([[(1,8,45),(2,46,5),(3,6,47),(4,48,7),(9,66,43),(10,44,67),(11,68,41),(12,42,65),(13,21,37),(14,38,22),(15,23,39),(16,40,24),(17,60,62),(18,63,57),(19,58,64),(20,61,59),(25,49,71),(26,72,50),(27,51,69),(28,70,52),(29,55,36),(30,33,56),(31,53,34),(32,35,54)], [(1,30,16),(2,31,13),(3,32,14),(4,29,15),(5,34,37),(6,35,38),(7,36,39),(8,33,40),(9,69,63),(10,70,64),(11,71,61),(12,72,62),(17,42,50),(18,43,51),(19,44,52),(20,41,49),(21,46,53),(22,47,54),(23,48,55),(24,45,56),(25,59,68),(26,60,65),(27,57,66),(28,58,67)], [(5,37,34),(6,35,38),(7,39,36),(8,33,40),(9,69,63),(10,64,70),(11,71,61),(12,62,72),(21,46,53),(22,54,47),(23,48,55),(24,56,45),(25,68,59),(26,60,65),(27,66,57),(28,58,67)], [(1,3),(2,4),(5,48),(6,45),(7,46),(8,47),(9,68),(10,65),(11,66),(12,67),(13,15),(14,16),(17,19),(18,20),(21,39),(22,40),(23,37),(24,38),(25,69),(26,70),(27,71),(28,72),(29,31),(30,32),(33,54),(34,55),(35,56),(36,53),(41,43),(42,44),(49,51),(50,52),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,18,3,20),(2,17,4,19),(5,60,7,58),(6,59,8,57),(9,22,11,24),(10,21,12,23),(13,42,15,44),(14,41,16,43),(25,33,27,35),(26,36,28,34),(29,52,31,50),(30,51,32,49),(37,65,39,67),(38,68,40,66),(45,63,47,61),(46,62,48,64),(53,72,55,70),(54,71,56,69)]])
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 | C12.85S32 |
| kernel | C12.85S32 | 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 C12.85S32 ►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] >;
C12.85S32 in GAP, Magma, Sage, TeX
C_{12}._{85}S_3^2 % in TeX
G:=Group("C12.85S3^2"); // 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