non-abelian, supersoluble, monomial
Aliases: C12⋊S3⋊4S3, C12.84(S32), He3⋊2(C4○D4), He3⋊3D4⋊3C2, He3⋊4D4⋊5C2, He3⋊3Q8⋊5C2, (C3×C12).22D6, C3⋊Dic3.1D6, C32⋊4Q8⋊4S3, C4.11(C32⋊D6), (C2×He3).3C23, C3.3(D12⋊S3), C32⋊2(D4⋊2S3), C32⋊1(Q8⋊3S3), C32⋊C12.1C22, (C4×He3).18C22, He3⋊3C4.14C22, C6.77(C2×S32), (C2×C3⋊S3).2D6, C32⋊C6⋊C4⋊2C2, C2.6(C2×C32⋊D6), (C4×He3⋊C2)⋊2C2, (C3×C6).3(C22×S3), (C2×C32⋊C6).1C22, (C2×He3⋊C2).11C22, SmallGroup(432,296)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.84(S32)
G = < a,b,c,d,e | a12=b3=d3=e2=1, c2=a6, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=b-1, dbd-1=a8b, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 927 in 156 conjugacy classes, 35 normal (23 characteristic)
C1, C2, C2 [×3], C3, C3 [×3], C4, C4 [×3], C22 [×3], S3 [×8], C6, C6 [×6], C2×C4 [×3], D4 [×3], Q8, C32 [×2], C32, Dic3 [×7], C12, C12 [×6], D6 [×7], C2×C6 [×3], C4○D4, C3×S3 [×6], C3⋊S3 [×2], C3×C6 [×2], C3×C6, Dic6 [×2], C4×S3 [×7], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4, C3×Q8, He3, C3×Dic3 [×5], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, S3×C6 [×5], C2×C3⋊S3 [×2], C4○D12, D4⋊2S3, Q8⋊3S3, C32⋊C6 [×2], He3⋊C2, C2×He3, S3×Dic3 [×2], C6.D6 [×2], D6⋊S3 [×2], C3⋊D12 [×2], C3×Dic6, S3×C12 [×3], C3×D12, C32⋊4Q8, C12⋊S3, C32⋊C12 [×2], He3⋊3C4, C4×He3, C2×C32⋊C6 [×2], C2×He3⋊C2, D12⋊5S3, D6.6D6, C32⋊C6⋊C4 [×2], He3⋊3D4 [×2], He3⋊3Q8, He3⋊4D4, C4×He3⋊C2, C12.84(S32)
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, D4⋊2S3, Q8⋊3S3, C2×S32, C32⋊D6, D12⋊S3, C2×C32⋊D6, C12.84(S32)
►On 72 points
(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 29 70)(2 30 71)(3 31 72)(4 32 61)(5 33 62)(6 34 63)(7 35 64)(8 36 65)(9 25 66)(10 26 67)(11 27 68)(12 28 69)(13 52 44)(14 53 45)(15 54 46)(16 55 47)(17 56 48)(18 57 37)(19 58 38)(20 59 39)(21 60 40)(22 49 41)(23 50 42)(24 51 43)
(1 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 62 19 68)(14 61 20 67)(15 72 21 66)(16 71 22 65)(17 70 23 64)(18 69 24 63)(25 54 31 60)(26 53 32 59)(27 52 33 58)(28 51 34 57)(29 50 35 56)(30 49 36 55)
(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)
(1 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(49 62)(50 61)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)
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), (1,29,70)(2,30,71)(3,31,72)(4,32,61)(5,33,62)(6,34,63)(7,35,64)(8,36,65)(9,25,66)(10,26,67)(11,27,68)(12,28,69)(13,52,44)(14,53,45)(15,54,46)(16,55,47)(17,56,48)(18,57,37)(19,58,38)(20,59,39)(21,60,40)(22,49,41)(23,50,42)(24,51,43), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,62,19,68)(14,61,20,67)(15,72,21,66)(16,71,22,65)(17,70,23,64)(18,69,24,63)(25,54,31,60)(26,53,32,59)(27,52,33,58)(28,51,34,57)(29,50,35,56)(30,49,36,55), (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)>;
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), (1,29,70)(2,30,71)(3,31,72)(4,32,61)(5,33,62)(6,34,63)(7,35,64)(8,36,65)(9,25,66)(10,26,67)(11,27,68)(12,28,69)(13,52,44)(14,53,45)(15,54,46)(16,55,47)(17,56,48)(18,57,37)(19,58,38)(20,59,39)(21,60,40)(22,49,41)(23,50,42)(24,51,43), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,62,19,68)(14,61,20,67)(15,72,21,66)(16,71,22,65)(17,70,23,64)(18,69,24,63)(25,54,31,60)(26,53,32,59)(27,52,33,58)(28,51,34,57)(29,50,35,56)(30,49,36,55), (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63) );
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)], [(1,29,70),(2,30,71),(3,31,72),(4,32,61),(5,33,62),(6,34,63),(7,35,64),(8,36,65),(9,25,66),(10,26,67),(11,27,68),(12,28,69),(13,52,44),(14,53,45),(15,54,46),(16,55,47),(17,56,48),(18,57,37),(19,58,38),(20,59,39),(21,60,40),(22,49,41),(23,50,42),(24,51,43)], [(1,42,7,48),(2,41,8,47),(3,40,9,46),(4,39,10,45),(5,38,11,44),(6,37,12,43),(13,62,19,68),(14,61,20,67),(15,72,21,66),(16,71,22,65),(17,70,23,64),(18,69,24,63),(25,54,31,60),(26,53,32,59),(27,52,33,58),(28,51,34,57),(29,50,35,56),(30,49,36,55)], [(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,48),(11,47),(12,46),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(49,62),(50,61),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63)])
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 | 2 | 2 | 12 | 12 | 12 | 12 | 18 | 18 | 36 | 36 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | C4○D4 | S32 | D4⋊2S3 | Q8⋊3S3 | C2×S32 | D12⋊S3 | C32⋊D6 | C2×C32⋊D6 | C12.84(S32) |
| kernel | C12.84(S32) | C32⋊C6⋊C4 | He3⋊3D4 | He3⋊3Q8 | He3⋊4D4 | C4×He3⋊C2 | C32⋊4Q8 | C12⋊S3 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | He3 | C12 | C32 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C12.84(S32) ►in GL6(𝔽13)
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0,5,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0] >;
C12.84(S32) in GAP, Magma, Sage, TeX
C_{12}._{84}(S_3^2) % in TeX
G:=Group("C12.84(S3^2)"); // GroupNames label
G:=SmallGroup(432,296);
// by ID
G=gap.SmallGroup(432,296);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^3=d^3=e^2=1,c^2=a^6,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a^8*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations