metabelian, supersoluble, monomial
Aliases: C12.95(S32), C33⋊5Q8⋊9C2, (C3×C12).171D6, C33⋊20(C4○D4), C33⋊9D4⋊13C2, C3⋊Dic3.47D6, C3⋊5(D6.D6), C32⋊14(C4○D12), C4.8(C32⋊4D6), (C32×C6).67C23, (C32×C12).74C22, C6.96(C2×S32), (C12×C3⋊S3)⋊2C2, (C4×C3⋊S3)⋊12S3, (C2×C3⋊S3).47D6, (C6×C3⋊S3).58C22, C2.5(C2×C32⋊4D6), (C3×C6).117(C22×S3), (C3×C3⋊Dic3).46C22, SmallGroup(432,689)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.95(S32)
G = < a,b,c,d,e | a12=b3=c2=d3=1, e2=a6, ab=ba, cac=eae-1=a5, ad=da, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a6c, ede-1=d-1 >
Subgroups: 1064 in 214 conjugacy classes, 47 normal (8 characteristic)
C1, C2, C2 [×3], C3 [×3], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×12], C6 [×3], C6 [×7], C2×C4 [×3], D4 [×3], Q8, C32 [×3], C32 [×4], Dic3 [×9], C12 [×3], C12 [×7], D6 [×9], C2×C6 [×3], C4○D4, C3×S3 [×12], C3⋊S3 [×3], C3×C6 [×3], C3×C6 [×4], Dic6 [×3], C4×S3 [×9], D12 [×3], C3⋊D4 [×6], C2×C12 [×3], C33, C3×Dic3 [×9], C3⋊Dic3 [×3], C3×C12 [×3], C3×C12 [×4], S3×C6 [×9], C2×C3⋊S3 [×3], C4○D12 [×3], C3×C3⋊S3 [×3], C32×C6, D6⋊S3 [×3], C3⋊D12 [×6], C32⋊2Q8 [×3], S3×C12 [×9], C4×C3⋊S3 [×3], C3×C3⋊Dic3 [×3], C32×C12, C6×C3⋊S3 [×3], D6.D6 [×3], C33⋊9D4 [×3], C33⋊5Q8, C12×C3⋊S3 [×3], C12.95(S32)
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12 [×3], C2×S32 [×3], C32⋊4D6, D6.D6 [×3], C2×C32⋊4D6, C12.95(S32)
►On 48 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)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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)
(1 23)(2 16)(3 21)(4 14)(5 19)(6 24)(7 17)(8 22)(9 15)(10 20)(11 13)(12 18)(25 47)(26 40)(27 45)(28 38)(29 43)(30 48)(31 41)(32 46)(33 39)(34 44)(35 37)(36 42)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 41 7 47)(2 46 8 40)(3 39 9 45)(4 44 10 38)(5 37 11 43)(6 42 12 48)(13 35 19 29)(14 28 20 34)(15 33 21 27)(16 26 22 32)(17 31 23 25)(18 36 24 30)
G:=sub<Sym(48)| (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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(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), (1,23)(2,16)(3,21)(4,14)(5,19)(6,24)(7,17)(8,22)(9,15)(10,20)(11,13)(12,18)(25,47)(26,40)(27,45)(28,38)(29,43)(30,48)(31,41)(32,46)(33,39)(34,44)(35,37)(36,42), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,41,7,47)(2,46,8,40)(3,39,9,45)(4,44,10,38)(5,37,11,43)(6,42,12,48)(13,35,19,29)(14,28,20,34)(15,33,21,27)(16,26,22,32)(17,31,23,25)(18,36,24,30)>;
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(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), (1,23)(2,16)(3,21)(4,14)(5,19)(6,24)(7,17)(8,22)(9,15)(10,20)(11,13)(12,18)(25,47)(26,40)(27,45)(28,38)(29,43)(30,48)(31,41)(32,46)(33,39)(34,44)(35,37)(36,42), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,41,7,47)(2,46,8,40)(3,39,9,45)(4,44,10,38)(5,37,11,43)(6,42,12,48)(13,35,19,29)(14,28,20,34)(15,33,21,27)(16,26,22,32)(17,31,23,25)(18,36,24,30) );
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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(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)], [(1,23),(2,16),(3,21),(4,14),(5,19),(6,24),(7,17),(8,22),(9,15),(10,20),(11,13),(12,18),(25,47),(26,40),(27,45),(28,38),(29,43),(30,48),(31,41),(32,46),(33,39),(34,44),(35,37),(36,42)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,41,7,47),(2,46,8,40),(3,39,9,45),(4,44,10,38),(5,37,11,43),(6,42,12,48),(13,35,19,29),(14,28,20,34),(15,33,21,27),(16,26,22,32),(17,31,23,25),(18,36,24,30)])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | ··· | 6H | 6I | ··· | 6N | 12A | ··· | 12F | 12G | ··· | 12P | 12Q | ··· | 12V |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 18 | ··· | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | ··· | 18 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | C2×S32 | C32⋊4D6 | D6.D6 | C2×C32⋊4D6 | C12.95(S32) |
| kernel | C12.95(S32) | C33⋊9D4 | C33⋊5Q8 | C12×C3⋊S3 | C4×C3⋊S3 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | C33 | C32 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 3 | 3 | 2 | 12 | 3 | 3 | 2 | 6 | 2 | 4 |
Matrix representation of C12.95(S32) ►in GL6(𝔽13)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 5 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,11,9,0,0,0,0,4,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,5,5] >;
C12.95(S32) in GAP, Magma, Sage, TeX
C_{12}._{95}(S_3^2) % in TeX
G:=Group("C12.95(S3^2)"); // GroupNames label
G:=SmallGroup(432,689);
// by ID
G=gap.SmallGroup(432,689);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,1124,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^3=1,e^2=a^6,a*b=b*a,c*a*c=e*a*e^-1=a^5,a*d=d*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^6*c,e*d*e^-1=d^-1>;
// generators/relations