metabelian, supersoluble, monomial
Aliases: C12.41S32, C32⋊9(S3×Q8), C33⋊11(C2×Q8), (C3×Dic6)⋊9S3, C33⋊C2⋊3Q8, C33⋊4Q8⋊7C2, Dic6⋊4(C3⋊S3), (C3×C12).144D6, C3⋊Dic3.36D6, C32⋊4Q8⋊11S3, (C3×Dic3).16D6, C3⋊3(Dic3.D6), (C32×Dic6)⋊13C2, (C32×C6).44C23, (C32×C12).46C22, C33⋊5C4.15C22, (C32×Dic3).16C22, C3⋊1(Q8×C3⋊S3), C6.54(C2×S32), C4.21(S3×C3⋊S3), C12.24(C2×C3⋊S3), C6.7(C22×C3⋊S3), C33⋊8(C2×C4).2C2, Dic3.4(C2×C3⋊S3), (C4×C33⋊C2).1C2, (C3×C32⋊4Q8)⋊10C2, (C3×C6).102(C22×S3), (C3×C3⋊Dic3).19C22, (C2×C33⋊C2).13C22, C2.11(C2×S3×C3⋊S3), SmallGroup(432,666)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊9(S3×Q8)
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, ac=ca, dad=a-1, ae=ea, af=fa, bc=cb, dbd=fbf-1=b-1, be=eb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 1648 in 292 conjugacy classes, 70 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×Q8, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, C2×C3⋊S3, S3×Q8, C33⋊C2, C32×C6, C6.D6, C32⋊2Q8, C3×Dic6, C3×Dic6, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, Q8×C32, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, C32×C12, C2×C33⋊C2, Dic3.D6, Q8×C3⋊S3, C33⋊8(C2×C4), C33⋊4Q8, C32×Dic6, C3×C32⋊4Q8, C4×C33⋊C2, C32⋊9(S3×Q8)
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, C22×S3, S32, C2×C3⋊S3, S3×Q8, C2×S32, C22×C3⋊S3, S3×C3⋊S3, Dic3.D6, Q8×C3⋊S3, C2×S3×C3⋊S3, C32⋊9(S3×Q8)
►On 72 points
Generators in S72
(1 29 21)(2 30 22)(3 31 23)(4 32 24)(5 51 35)(6 52 36)(7 49 33)(8 50 34)(9 43 69)(10 44 70)(11 41 71)(12 42 72)(13 60 38)(14 57 39)(15 58 40)(16 59 37)(17 53 45)(18 54 46)(19 55 47)(20 56 48)(25 68 61)(26 65 62)(27 66 63)(28 67 64)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 43 69)(10 44 70)(11 41 71)(12 42 72)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 53 45)(18 54 46)(19 55 47)(20 56 48)(25 68 61)(26 65 62)(27 66 63)(28 67 64)
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 3)(2 4)(5 40)(6 37)(7 38)(8 39)(9 25)(10 26)(11 27)(12 28)(13 33)(14 34)(15 35)(16 36)(17 19)(18 20)(21 31)(22 32)(23 29)(24 30)(41 63)(42 64)(43 61)(44 62)(45 55)(46 56)(47 53)(48 54)(49 60)(50 57)(51 58)(52 59)(65 70)(66 71)(67 72)(68 69)
(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 17 3 19)(2 20 4 18)(5 64 7 62)(6 63 8 61)(9 59 11 57)(10 58 12 60)(13 70 15 72)(14 69 16 71)(21 45 23 47)(22 48 24 46)(25 52 27 50)(26 51 28 49)(29 53 31 55)(30 56 32 54)(33 65 35 67)(34 68 36 66)(37 41 39 43)(38 44 40 42)
G:=sub<Sym(72)| (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,51,35)(6,52,36)(7,49,33)(8,50,34)(9,43,69)(10,44,70)(11,41,71)(12,42,72)(13,60,38)(14,57,39)(15,58,40)(16,59,37)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(25,68,61)(26,65,62)(27,66,63)(28,67,64), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,43,69)(10,44,70)(11,41,71)(12,42,72)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(25,68,61)(26,65,62)(27,66,63)(28,67,64), (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,3)(2,4)(5,40)(6,37)(7,38)(8,39)(9,25)(10,26)(11,27)(12,28)(13,33)(14,34)(15,35)(16,36)(17,19)(18,20)(21,31)(22,32)(23,29)(24,30)(41,63)(42,64)(43,61)(44,62)(45,55)(46,56)(47,53)(48,54)(49,60)(50,57)(51,58)(52,59)(65,70)(66,71)(67,72)(68,69), (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), (1,17,3,19)(2,20,4,18)(5,64,7,62)(6,63,8,61)(9,59,11,57)(10,58,12,60)(13,70,15,72)(14,69,16,71)(21,45,23,47)(22,48,24,46)(25,52,27,50)(26,51,28,49)(29,53,31,55)(30,56,32,54)(33,65,35,67)(34,68,36,66)(37,41,39,43)(38,44,40,42)>;
G:=Group( (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,51,35)(6,52,36)(7,49,33)(8,50,34)(9,43,69)(10,44,70)(11,41,71)(12,42,72)(13,60,38)(14,57,39)(15,58,40)(16,59,37)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(25,68,61)(26,65,62)(27,66,63)(28,67,64), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,43,69)(10,44,70)(11,41,71)(12,42,72)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(25,68,61)(26,65,62)(27,66,63)(28,67,64), (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,3)(2,4)(5,40)(6,37)(7,38)(8,39)(9,25)(10,26)(11,27)(12,28)(13,33)(14,34)(15,35)(16,36)(17,19)(18,20)(21,31)(22,32)(23,29)(24,30)(41,63)(42,64)(43,61)(44,62)(45,55)(46,56)(47,53)(48,54)(49,60)(50,57)(51,58)(52,59)(65,70)(66,71)(67,72)(68,69), (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), (1,17,3,19)(2,20,4,18)(5,64,7,62)(6,63,8,61)(9,59,11,57)(10,58,12,60)(13,70,15,72)(14,69,16,71)(21,45,23,47)(22,48,24,46)(25,52,27,50)(26,51,28,49)(29,53,31,55)(30,56,32,54)(33,65,35,67)(34,68,36,66)(37,41,39,43)(38,44,40,42) );
G=PermutationGroup([[(1,29,21),(2,30,22),(3,31,23),(4,32,24),(5,51,35),(6,52,36),(7,49,33),(8,50,34),(9,43,69),(10,44,70),(11,41,71),(12,42,72),(13,60,38),(14,57,39),(15,58,40),(16,59,37),(17,53,45),(18,54,46),(19,55,47),(20,56,48),(25,68,61),(26,65,62),(27,66,63),(28,67,64)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,51),(6,36,52),(7,33,49),(8,34,50),(9,43,69),(10,44,70),(11,41,71),(12,42,72),(13,38,60),(14,39,57),(15,40,58),(16,37,59),(17,53,45),(18,54,46),(19,55,47),(20,56,48),(25,68,61),(26,65,62),(27,66,63),(28,67,64)], [(1,40,7),(2,37,8),(3,38,5),(4,39,6),(9,66,46),(10,67,47),(11,68,48),(12,65,45),(13,51,31),(14,52,32),(15,49,29),(16,50,30),(17,42,62),(18,43,63),(19,44,64),(20,41,61),(21,58,33),(22,59,34),(23,60,35),(24,57,36),(25,56,71),(26,53,72),(27,54,69),(28,55,70)], [(1,3),(2,4),(5,40),(6,37),(7,38),(8,39),(9,25),(10,26),(11,27),(12,28),(13,33),(14,34),(15,35),(16,36),(17,19),(18,20),(21,31),(22,32),(23,29),(24,30),(41,63),(42,64),(43,61),(44,62),(45,55),(46,56),(47,53),(48,54),(49,60),(50,57),(51,58),(52,59),(65,70),(66,71),(67,72),(68,69)], [(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)], [(1,17,3,19),(2,20,4,18),(5,64,7,62),(6,63,8,61),(9,59,11,57),(10,58,12,60),(13,70,15,72),(14,69,16,71),(21,45,23,47),(22,48,24,46),(25,52,27,50),(26,51,28,49),(29,53,31,55),(30,56,32,54),(33,65,35,67),(34,68,36,66),(37,41,39,43),(38,44,40,42)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 12A | ··· | 12M | 12N | ··· | 12U | 12V | 12W |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 27 | 27 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 6 | 6 | 18 | 18 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 36 | 36 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D6 | S32 | S3×Q8 | C2×S32 | Dic3.D6 |
| kernel | C32⋊9(S3×Q8) | C33⋊8(C2×C4) | C33⋊4Q8 | C32×Dic6 | C3×C32⋊4Q8 | C4×C33⋊C2 | C3×Dic6 | C32⋊4Q8 | C33⋊C2 | C3×Dic3 | C3⋊Dic3 | C3×C12 | C12 | C32 | C6 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 2 | 8 | 2 | 5 | 4 | 5 | 4 | 8 |
Matrix representation of C32⋊9(S3×Q8) ►in GL8(𝔽13)
| 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 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 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 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 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 2 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,7,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,12,11,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
C32⋊9(S3×Q8) in GAP, Magma, Sage, TeX
C_3^2\rtimes_9(S_3\times Q_8)
% in TeX
G:=Group("C3^2:9(S3xQ8)"); // GroupNames label
G:=SmallGroup(432,666);
// by ID
G=gap.SmallGroup(432,666);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,135,58,571,2028,14118]);
// 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,a*c=c*a,d*a*d=a^-1,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d=f*b*f^-1=b^-1,b*e=e*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations