non-abelian, supersoluble, monomial
Aliases: C12.23(S32), He3⋊4(C4○D4), He3⋊3D4⋊4C2, He3⋊5D4⋊3C2, He3⋊3Q8⋊7C2, (C3×C12).24D6, C3⋊Dic3.3D6, C32⋊4Q8⋊6S3, C4.7(C32⋊D6), C32⋊3(C4○D12), (C2×He3).6C23, C3.3(D6.6D6), C32⋊2(Q8⋊3S3), C32⋊C12.3C22, (C4×He3).20C22, (C4×C3⋊S3)⋊2S3, C6.80(C2×S32), (C2×C3⋊S3).7D6, (C4×C32⋊C6)⋊3C2, C2.9(C2×C32⋊D6), He3⋊(C2×C4)⋊1C2, (C3×C6).6(C22×S3), (C2×C32⋊C6).7C22, (C2×He3⋊C2).4C22, SmallGroup(432,299)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.(S32)
G = < a,b,c,d,e | a12=b3=d3=e2=1, c2=a6, ab=ba, cac-1=a-1, ad=da, eae=a5, cbc-1=b-1, dbd-1=a4b, be=eb, cd=dc, ece=a6c, ede=d-1 >
Subgroups: 943 in 156 conjugacy classes, 35 normal (25 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 [×6], C12, C12 [×6], D6 [×8], C2×C6 [×3], C4○D4, C3×S3 [×7], C3⋊S3, C3×C6 [×2], C3×C6, Dic6 [×2], C4×S3 [×6], D12 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4, C3×Q8, He3, C3×Dic3 [×3], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, S3×C6 [×7], C2×C3⋊S3, C4○D12, D4⋊2S3, Q8⋊3S3, C32⋊C6, He3⋊C2 [×2], C2×He3, S3×Dic3 [×4], D6⋊S3 [×2], C3⋊D12 [×2], C3×Dic6, S3×C12, C3×D12 [×3], C32⋊4Q8, C4×C3⋊S3, C32⋊C12, C32⋊C12 [×2], C4×He3, C2×C32⋊C6, C2×He3⋊C2 [×2], D12⋊5S3, D12⋊S3, He3⋊(C2×C4) [×2], He3⋊3D4 [×2], He3⋊3Q8, C4×C32⋊C6, He3⋊5D4, C12.(S32)
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, C4○D12, Q8⋊3S3, C2×S32, C32⋊D6, D6.6D6, C2×C32⋊D6, C12.(S32)
►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)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(1 46 7 40)(2 45 8 39)(3 44 9 38)(4 43 10 37)(5 42 11 48)(6 41 12 47)(13 66 19 72)(14 65 20 71)(15 64 21 70)(16 63 22 69)(17 62 23 68)(18 61 24 67)(25 51 31 57)(26 50 32 56)(27 49 33 55)(28 60 34 54)(29 59 35 53)(30 58 36 52)
(1 21 26)(2 22 27)(3 23 28)(4 24 29)(5 13 30)(6 14 31)(7 15 32)(8 16 33)(9 17 34)(10 18 35)(11 19 36)(12 20 25)(37 61 53)(38 62 54)(39 63 55)(40 64 56)(41 65 57)(42 66 58)(43 67 59)(44 68 60)(45 69 49)(46 70 50)(47 71 51)(48 72 52)
(1 37)(2 42)(3 47)(4 40)(5 45)(6 38)(7 43)(8 48)(9 41)(10 46)(11 39)(12 44)(13 49)(14 54)(15 59)(16 52)(17 57)(18 50)(19 55)(20 60)(21 53)(22 58)(23 51)(24 56)(25 68)(26 61)(27 66)(28 71)(29 64)(30 69)(31 62)(32 67)(33 72)(34 65)(35 70)(36 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), (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,46,7,40)(2,45,8,39)(3,44,9,38)(4,43,10,37)(5,42,11,48)(6,41,12,47)(13,66,19,72)(14,65,20,71)(15,64,21,70)(16,63,22,69)(17,62,23,68)(18,61,24,67)(25,51,31,57)(26,50,32,56)(27,49,33,55)(28,60,34,54)(29,59,35,53)(30,58,36,52), (1,21,26)(2,22,27)(3,23,28)(4,24,29)(5,13,30)(6,14,31)(7,15,32)(8,16,33)(9,17,34)(10,18,35)(11,19,36)(12,20,25)(37,61,53)(38,62,54)(39,63,55)(40,64,56)(41,65,57)(42,66,58)(43,67,59)(44,68,60)(45,69,49)(46,70,50)(47,71,51)(48,72,52), (1,37)(2,42)(3,47)(4,40)(5,45)(6,38)(7,43)(8,48)(9,41)(10,46)(11,39)(12,44)(13,49)(14,54)(15,59)(16,52)(17,57)(18,50)(19,55)(20,60)(21,53)(22,58)(23,51)(24,56)(25,68)(26,61)(27,66)(28,71)(29,64)(30,69)(31,62)(32,67)(33,72)(34,65)(35,70)(36,63)>;
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), (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,46,7,40)(2,45,8,39)(3,44,9,38)(4,43,10,37)(5,42,11,48)(6,41,12,47)(13,66,19,72)(14,65,20,71)(15,64,21,70)(16,63,22,69)(17,62,23,68)(18,61,24,67)(25,51,31,57)(26,50,32,56)(27,49,33,55)(28,60,34,54)(29,59,35,53)(30,58,36,52), (1,21,26)(2,22,27)(3,23,28)(4,24,29)(5,13,30)(6,14,31)(7,15,32)(8,16,33)(9,17,34)(10,18,35)(11,19,36)(12,20,25)(37,61,53)(38,62,54)(39,63,55)(40,64,56)(41,65,57)(42,66,58)(43,67,59)(44,68,60)(45,69,49)(46,70,50)(47,71,51)(48,72,52), (1,37)(2,42)(3,47)(4,40)(5,45)(6,38)(7,43)(8,48)(9,41)(10,46)(11,39)(12,44)(13,49)(14,54)(15,59)(16,52)(17,57)(18,50)(19,55)(20,60)(21,53)(22,58)(23,51)(24,56)(25,68)(26,61)(27,66)(28,71)(29,64)(30,69)(31,62)(32,67)(33,72)(34,65)(35,70)(36,63) );
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)], [(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72)], [(1,46,7,40),(2,45,8,39),(3,44,9,38),(4,43,10,37),(5,42,11,48),(6,41,12,47),(13,66,19,72),(14,65,20,71),(15,64,21,70),(16,63,22,69),(17,62,23,68),(18,61,24,67),(25,51,31,57),(26,50,32,56),(27,49,33,55),(28,60,34,54),(29,59,35,53),(30,58,36,52)], [(1,21,26),(2,22,27),(3,23,28),(4,24,29),(5,13,30),(6,14,31),(7,15,32),(8,16,33),(9,17,34),(10,18,35),(11,19,36),(12,20,25),(37,61,53),(38,62,54),(39,63,55),(40,64,56),(41,65,57),(42,66,58),(43,67,59),(44,68,60),(45,69,49),(46,70,50),(47,71,51),(48,72,52)], [(1,37),(2,42),(3,47),(4,40),(5,45),(6,38),(7,43),(8,48),(9,41),(10,46),(11,39),(12,44),(13,49),(14,54),(15,59),(16,52),(17,57),(18,50),(19,55),(20,60),(21,53),(22,58),(23,51),(24,56),(25,68),(26,61),(27,66),(28,71),(29,64),(30,69),(31,62),(32,67),(33,72),(34,65),(35,70),(36,63)])
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 | 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.(S32) | S3 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | Q8⋊3S3 | C2×S32 | D6.6D6 | C32⋊D6 | C2×C32⋊D6 |
| kernel | C12.(S32) | He3⋊(C2×C4) | He3⋊3D4 | He3⋊3Q8 | C4×C32⋊C6 | He3⋊5D4 | C1 | C32⋊4Q8 | C4×C3⋊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 | 3 | 2 | 1 | 2 | 4 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C12.(S32) ►in GL10(𝔽13)
| 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 | 9 | 0 | 0 | 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 | 4 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 | 0 | 0 | 0 | 3 |
| 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 9 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 3 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 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 | 1 | 12 | 12 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 8 | 0 | 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 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 9 | 2 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 11 | 4 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
G:=sub<GL(10,GF(13))| [10,7,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,9,0,0,0,0,0,9,0,0,4,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,10,0,0,0,0,0,3,9,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3],[0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8,0,1,1,0,12,0,0,0,0,0,8,1,0,1,12,0,0,0,0,0,0,1,0,0,0],[0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,8,1,0,1,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[11,9,2,4,0,0,0,0,0,0,4,2,9,11,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,9,11,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,1,1,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,1,0,1,12] >;
C12.(S32) in GAP, Magma, Sage, TeX
C_{12}.(S_3^2) % in TeX
G:=Group("C12.(S3^2)"); // GroupNames label
G:=SmallGroup(432,299);
// by ID
G=gap.SmallGroup(432,299);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,135,58,571,4037,537,14118,7069]);
// 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=a^-1,a*d=d*a,e*a*e=a^5,c*b*c^-1=b^-1,d*b*d^-1=a^4*b,b*e=e*b,c*d=d*c,e*c*e=a^6*c,e*d*e=d^-1>;
// generators/relations