metabelian, supersoluble, monomial
Aliases: C12.43(S32), C12⋊S3⋊12S3, (C3×C12).148D6, C33⋊19(C4○D4), C33⋊9D4⋊11C2, C3⋊Dic3.21D6, C3⋊5(D12⋊5S3), C3⋊5(D12⋊S3), C32⋊4Q8⋊13S3, C3⋊5(D6.6D6), C32⋊13(C4○D12), C32⋊8(Q8⋊3S3), C4.5(C32⋊4D6), (C32×C6).66C23, C32⋊14(D4⋊2S3), (C32×C12).50C22, (C4×C3⋊S3)⋊9S3, C6.95(C2×S32), (C12×C3⋊S3)⋊8C2, (C2×C3⋊S3).46D6, C33⋊9(C2×C4)⋊4C2, (C3×C12⋊S3)⋊12C2, (C6×C3⋊S3).30C22, (C3×C32⋊4Q8)⋊13C2, C2.4(C2×C32⋊4D6), (C3×C6).116(C22×S3), (C3×C3⋊Dic3).24C22, SmallGroup(432,688)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9(C2×C4) — (C3×C12).148D6 |
Generators and relations for (C3×C12).148D6
G = < a,b,c,d | a3=b12=1, c6=d2=b6, ab=ba, cac-1=a-1, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >
Subgroups: 1064 in 210 conjugacy classes, 47 normal (35 characteristic)
C1, C2, C2 [×3], C3 [×3], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×10], 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 [×10], C3⋊S3 [×3], C3×C6 [×3], C3×C6 [×4], Dic6 [×3], C4×S3 [×7], D12 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4, C3×Q8, C33, C3×Dic3 [×9], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12 [×3], C3×C12 [×4], S3×C6 [×9], C2×C3⋊S3, C2×C3⋊S3 [×2], C4○D12, D4⋊2S3, Q8⋊3S3, C3×C3⋊S3 [×3], C32×C6, S3×Dic3 [×4], C6.D6 [×2], D6⋊S3 [×2], C3⋊D12 [×4], C3×Dic6 [×3], S3×C12 [×3], C3×D12 [×3], C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C3×C3⋊Dic3, C3×C3⋊Dic3 [×2], C32×C12, C6×C3⋊S3, C6×C3⋊S3 [×2], D12⋊5S3, D12⋊S3, D6.6D6, C33⋊9(C2×C4) [×2], C33⋊9D4 [×2], C3×C32⋊4Q8, C12×C3⋊S3, C3×C12⋊S3, (C3×C12).148D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12, D4⋊2S3, Q8⋊3S3, C2×S32 [×3], C32⋊4D6, D12⋊5S3, D12⋊S3, D6.6D6, C2×C32⋊4D6, (C3×C12).148D6
(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 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 18 3 16 5 14 7 24 9 22 11 20)(2 23 4 21 6 19 8 17 10 15 12 13)(25 43 35 45 33 47 31 37 29 39 27 41)(26 48 36 38 34 40 32 42 30 44 28 46)
(1 41 7 47)(2 40 8 46)(3 39 9 45)(4 38 10 44)(5 37 11 43)(6 48 12 42)(13 26 19 32)(14 25 20 31)(15 36 21 30)(16 35 22 29)(17 34 23 28)(18 33 24 27)
G:=sub<Sym(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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,18,3,16,5,14,7,24,9,22,11,20)(2,23,4,21,6,19,8,17,10,15,12,13)(25,43,35,45,33,47,31,37,29,39,27,41)(26,48,36,38,34,40,32,42,30,44,28,46), (1,41,7,47)(2,40,8,46)(3,39,9,45)(4,38,10,44)(5,37,11,43)(6,48,12,42)(13,26,19,32)(14,25,20,31)(15,36,21,30)(16,35,22,29)(17,34,23,28)(18,33,24,27)>;
G:=Group( (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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,18,3,16,5,14,7,24,9,22,11,20)(2,23,4,21,6,19,8,17,10,15,12,13)(25,43,35,45,33,47,31,37,29,39,27,41)(26,48,36,38,34,40,32,42,30,44,28,46), (1,41,7,47)(2,40,8,46)(3,39,9,45)(4,38,10,44)(5,37,11,43)(6,48,12,42)(13,26,19,32)(14,25,20,31)(15,36,21,30)(16,35,22,29)(17,34,23,28)(18,33,24,27) );
G=PermutationGroup([(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)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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)], [(1,18,3,16,5,14,7,24,9,22,11,20),(2,23,4,21,6,19,8,17,10,15,12,13),(25,43,35,45,33,47,31,37,29,39,27,41),(26,48,36,38,34,40,32,42,30,44,28,46)], [(1,41,7,47),(2,40,8,46),(3,39,9,45),(4,38,10,44),(5,37,11,43),(6,48,12,42),(13,26,19,32),(14,25,20,31),(15,36,21,30),(16,35,22,29),(17,34,23,28),(18,33,24,27)])
48 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | ··· | 6H | 6I | 6J | 6K | 6L | 12A | 12B | 12C | ··· | 12N | 12O | 12P | 12Q | 12R |
order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
size | 1 | 1 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 9 | 9 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 36 | 36 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 36 | 36 |
48 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | ||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | D4⋊2S3 | Q8⋊3S3 | C2×S32 | C32⋊4D6 | D12⋊5S3 | D12⋊S3 | D6.6D6 | C2×C32⋊4D6 | (C3×C12).148D6 |
kernel | (C3×C12).148D6 | C33⋊9(C2×C4) | C33⋊9D4 | C3×C32⋊4Q8 | C12×C3⋊S3 | C3×C12⋊S3 | C32⋊4Q8 | C4×C3⋊S3 | C12⋊S3 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | C33 | C32 | C12 | C32 | C32 | C6 | C4 | C3 | C3 | C3 | C2 | C1 |
# reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 2 | 4 | 3 | 1 | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 4 |
Matrix representation of (C3×C12).148D6 ►in GL6(𝔽13)
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 |
0 | 1 | 0 | 0 | 0 | 0 |
12 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 3 | 7 |
0 | 0 | 0 | 0 | 6 | 10 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 12 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 8 | 5 |
0 | 0 | 0 | 0 | 8 | 0 |
0 | 12 | 0 | 0 | 0 | 0 |
12 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 5 |
0 | 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(6,GF(13))| [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],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,8,0,0,0,0,5,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0] >;
(C3×C12).148D6 in GAP, Magma, Sage, TeX
(C_3\times C_{12})._{148}D_6
% in TeX
G:=Group("(C3xC12).148D6");
// GroupNames label
G:=SmallGroup(432,688);
// by ID
G=gap.SmallGroup(432,688);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,1124,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=1,c^6=d^2=b^6,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations