non-abelian, supersoluble, monomial
Aliases: C12.22S32, C12⋊S3⋊3S3, He3⋊1(C4○D4), He3⋊2D4⋊4C2, He3⋊4D4⋊4C2, He3⋊4Q8⋊4C2, (C3×C12).21D6, C3⋊Dic3.6D6, C4.6(C32⋊D6), C32⋊1(C4○D12), (C2×He3).2C23, C3.3(D12⋊5S3), C32⋊1(D4⋊2S3), C32⋊C12.9C22, (C4×He3).17C22, He3⋊3C4.5C22, (C4×C3⋊S3)⋊1S3, C6.76(C2×S32), (C2×C3⋊S3).1D6, C6.S32⋊1C2, (C4×C32⋊C6)⋊2C2, C2.5(C2×C32⋊D6), (C3×C6).2(C22×S3), (C2×C32⋊C6).5C22, SmallGroup(432,295)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊S3⋊S3
G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, cac=a-1, ad=da, eae=a5, cbc=b-1, dbd-1=a8b, be=eb, cd=dc, ece=a6c, ede=d-1 >
Subgroups: 911 in 152 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C4○D12, D4⋊2S3, Q8⋊3S3, C32⋊C6, C2×He3, S3×Dic3, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, C32⋊C12, He3⋊3C4, C4×He3, C2×C32⋊C6, C2×C32⋊C6, D12⋊S3, D6.6D6, C6.S32, He3⋊2D4, C4×C32⋊C6, He3⋊4D4, He3⋊4Q8, C12⋊S3⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, C2×S32, C32⋊D6, D12⋊5S3, C2×C32⋊D6, C12⋊S3⋊S3
►On 72 points
Generators in S72
(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)
(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 19)(14 18)(15 17)(20 24)(21 23)(25 35)(26 34)(27 33)(28 32)(29 31)(37 47)(38 46)(39 45)(40 44)(41 43)(49 55)(50 54)(51 53)(56 60)(57 59)(62 72)(63 71)(64 70)(65 69)(66 68)
(1 42 61)(2 43 62)(3 44 63)(4 45 64)(5 46 65)(6 47 66)(7 48 67)(8 37 68)(9 38 69)(10 39 70)(11 40 71)(12 41 72)(13 49 27)(14 50 28)(15 51 29)(16 52 30)(17 53 31)(18 54 32)(19 55 33)(20 56 34)(21 57 35)(22 58 36)(23 59 25)(24 60 26)
(1 55)(2 60)(3 53)(4 58)(5 51)(6 56)(7 49)(8 54)(9 59)(10 52)(11 57)(12 50)(13 48)(14 41)(15 46)(16 39)(17 44)(18 37)(19 42)(20 47)(21 40)(22 45)(23 38)(24 43)(25 69)(26 62)(27 67)(28 72)(29 65)(30 70)(31 63)(32 68)(33 61)(34 66)(35 71)(36 64)
G:=sub<Sym(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), (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (2,12)(3,11)(4,10)(5,9)(6,8)(13,19)(14,18)(15,17)(20,24)(21,23)(25,35)(26,34)(27,33)(28,32)(29,31)(37,47)(38,46)(39,45)(40,44)(41,43)(49,55)(50,54)(51,53)(56,60)(57,59)(62,72)(63,71)(64,70)(65,69)(66,68), (1,42,61)(2,43,62)(3,44,63)(4,45,64)(5,46,65)(6,47,66)(7,48,67)(8,37,68)(9,38,69)(10,39,70)(11,40,71)(12,41,72)(13,49,27)(14,50,28)(15,51,29)(16,52,30)(17,53,31)(18,54,32)(19,55,33)(20,56,34)(21,57,35)(22,58,36)(23,59,25)(24,60,26), (1,55)(2,60)(3,53)(4,58)(5,51)(6,56)(7,49)(8,54)(9,59)(10,52)(11,57)(12,50)(13,48)(14,41)(15,46)(16,39)(17,44)(18,37)(19,42)(20,47)(21,40)(22,45)(23,38)(24,43)(25,69)(26,62)(27,67)(28,72)(29,65)(30,70)(31,63)(32,68)(33,61)(34,66)(35,71)(36,64)>;
G:=Group( (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), (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (2,12)(3,11)(4,10)(5,9)(6,8)(13,19)(14,18)(15,17)(20,24)(21,23)(25,35)(26,34)(27,33)(28,32)(29,31)(37,47)(38,46)(39,45)(40,44)(41,43)(49,55)(50,54)(51,53)(56,60)(57,59)(62,72)(63,71)(64,70)(65,69)(66,68), (1,42,61)(2,43,62)(3,44,63)(4,45,64)(5,46,65)(6,47,66)(7,48,67)(8,37,68)(9,38,69)(10,39,70)(11,40,71)(12,41,72)(13,49,27)(14,50,28)(15,51,29)(16,52,30)(17,53,31)(18,54,32)(19,55,33)(20,56,34)(21,57,35)(22,58,36)(23,59,25)(24,60,26), (1,55)(2,60)(3,53)(4,58)(5,51)(6,56)(7,49)(8,54)(9,59)(10,52)(11,57)(12,50)(13,48)(14,41)(15,46)(16,39)(17,44)(18,37)(19,42)(20,47)(21,40)(22,45)(23,38)(24,43)(25,69)(26,62)(27,67)(28,72)(29,65)(30,70)(31,63)(32,68)(33,61)(34,66)(35,71)(36,64) );
G=PermutationGroup([[(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)], [(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(61,65,69),(62,66,70),(63,67,71),(64,68,72)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,19),(14,18),(15,17),(20,24),(21,23),(25,35),(26,34),(27,33),(28,32),(29,31),(37,47),(38,46),(39,45),(40,44),(41,43),(49,55),(50,54),(51,53),(56,60),(57,59),(62,72),(63,71),(64,70),(65,69),(66,68)], [(1,42,61),(2,43,62),(3,44,63),(4,45,64),(5,46,65),(6,47,66),(7,48,67),(8,37,68),(9,38,69),(10,39,70),(11,40,71),(12,41,72),(13,49,27),(14,50,28),(15,51,29),(16,52,30),(17,53,31),(18,54,32),(19,55,33),(20,56,34),(21,57,35),(22,58,36),(23,59,25),(24,60,26)], [(1,55),(2,60),(3,53),(4,58),(5,51),(6,56),(7,49),(8,54),(9,59),(10,52),(11,57),(12,50),(13,48),(14,41),(15,46),(16,39),(17,44),(18,37),(19,42),(20,47),(21,40),(22,45),(23,38),(24,43),(25,69),(26,62),(27,67),(28,72),(29,65),(30,70),(31,63),(32,68),(33,61),(34,66),(35,71),(36,64)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 12 | 12 |
| size | 1 | 1 | 18 | 18 | 18 | 2 | 6 | 6 | 12 | 2 | 9 | 9 | 18 | 18 | 2 | 6 | 6 | 12 | 18 | 18 | 36 | 36 | 4 | 6 | 6 | 12 | 12 | 12 | 18 | 18 | 36 | 36 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C12⋊S3⋊S3 | S3 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | D4⋊2S3 | C2×S32 | D12⋊5S3 | C32⋊D6 | C2×C32⋊D6 |
| kernel | C12⋊S3⋊S3 | C6.S32 | He3⋊2D4 | C4×C32⋊C6 | He3⋊4D4 | He3⋊4Q8 | C1 | C4×C3⋊S3 | C12⋊S3 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | He3 | C32 | C12 | C32 | C6 | C3 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 4 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C12⋊S3⋊S3 ►in GL10(𝔽13)
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 12 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 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 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 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 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 12 | 12 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 11 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 | 0 |
| 12 | 11 | 9 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 8 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 5 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 4 | 11 | 12 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 12 | 0 | 0 |
G:=sub<GL(10,GF(13))| [10,6,0,0,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,0,10,6,0,0,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,0,12,12,12,0,12,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12],[3,0,8,0,0,0,0,0,0,0,0,3,0,8,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,11,12,12,12,12,0,0,0,0,1,12,0,0,0,0],[12,2,9,8,0,0,0,0,0,0,11,1,5,4,0,0,0,0,0,0,9,8,1,11,0,0,0,0,0,0,5,4,2,12,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0] >;
C12⋊S3⋊S3 in GAP, Magma, Sage, TeX
C_{12}\rtimes S_3\rtimes S_3 % in TeX
G:=Group("C12:S3:S3"); // GroupNames label
G:=SmallGroup(432,295);
// by ID
G=gap.SmallGroup(432,295);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^3=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,e*a*e=a^5,c*b*c=b^-1,d*b*d^-1=a^8*b,b*e=e*b,c*d=d*c,e*c*e=a^6*c,e*d*e=d^-1>;
// generators/relations