metabelian, supersoluble, monomial
Aliases: D12.25D6, Dic6.25D6, C32⋊42- 1+4, (S3×Q8)⋊7S3, Q8.18S32, Q8⋊3S3⋊7S3, C3⋊4(Q8○D12), (C4×S3).16D6, (C3×Q8).46D6, (S3×Dic6)⋊11C2, D6.D6⋊8C2, (C3×C6).22C24, C6.22(S3×C23), D12⋊S3⋊11C2, D12⋊5S3⋊11C2, (S3×C6).24C23, C12.34(C22×S3), (C3×C12).34C23, D6.22(C22×S3), (S3×C12).33C22, C3⋊3(Q8.15D6), D6⋊S3.9C22, C3⋊D12.3C22, (C3×D12).30C22, C3⋊Dic3.24C23, (S3×Dic3).4C22, Dic3.11(C22×S3), (C3×Dic6).30C22, (C3×Dic3).15C23, C32⋊2Q8.10C22, (Q8×C32).21C22, C32⋊4Q8.22C22, (C3×S3×Q8)⋊7C2, C4.34(C2×S32), (Q8×C3⋊S3)⋊6C2, C2.24(C22×S32), (C3×Q8⋊3S3)⋊7C2, (C4×C3⋊S3).44C22, (C2×C3⋊S3).46C23, SmallGroup(288,963)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.25D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, bd=db, dcd-1=a6c5 >
Subgroups: 1026 in 311 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4 [×3], C4 [×7], C22 [×5], S3 [×8], C6 [×2], C6 [×5], C2×C4 [×15], D4 [×10], Q8, Q8 [×9], C32, Dic3, Dic3 [×3], Dic3 [×9], C12 [×6], C12 [×7], D6, D6 [×3], D6 [×3], C2×C6 [×4], C2×Q8 [×5], C4○D4 [×10], C3×S3 [×4], C3⋊S3, C3×C6, Dic6 [×3], Dic6 [×15], C4×S3 [×6], C4×S3 [×15], D12 [×3], D12 [×4], C2×Dic3 [×6], C3⋊D4 [×10], C2×C12 [×6], C3×D4 [×3], C3×Q8 [×2], C3×Q8 [×4], 2- 1+4, C3×Dic3, C3×Dic3 [×3], C3⋊Dic3 [×3], C3×C12 [×3], S3×C6, S3×C6 [×3], C2×C3⋊S3, C2×Dic6 [×3], C4○D12 [×9], D4⋊2S3 [×6], S3×Q8, S3×Q8 [×6], Q8⋊3S3, Q8⋊3S3 [×3], C6×Q8, C3×C4○D4, S3×Dic3 [×6], D6⋊S3 [×3], C3⋊D12, C3⋊D12 [×3], C32⋊2Q8 [×3], C3×Dic6 [×3], S3×C12 [×6], C3×D12 [×3], C32⋊4Q8 [×3], C4×C3⋊S3 [×3], Q8×C32, Q8.15D6, Q8○D12, S3×Dic6 [×3], D12⋊5S3 [×3], D12⋊S3 [×3], D6.D6 [×3], C3×S3×Q8, C3×Q8⋊3S3, Q8×C3⋊S3, D12.25D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2- 1+4, S32, S3×C23 [×2], C2×S32 [×3], Q8.15D6, Q8○D12, C22×S32, D12.25D6
►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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 48)(13 28)(14 27)(15 26)(16 25)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)
(1 21 3 23 5 13 7 15 9 17 11 19)(2 16 4 18 6 20 8 22 10 24 12 14)(25 38 35 48 33 46 31 44 29 42 27 40)(26 45 36 43 34 41 32 39 30 37 28 47)
(1 32 7 26)(2 33 8 27)(3 34 9 28)(4 35 10 29)(5 36 11 30)(6 25 12 31)(13 45 19 39)(14 46 20 40)(15 47 21 41)(16 48 22 42)(17 37 23 43)(18 38 24 44)
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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,48)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29), (1,21,3,23,5,13,7,15,9,17,11,19)(2,16,4,18,6,20,8,22,10,24,12,14)(25,38,35,48,33,46,31,44,29,42,27,40)(26,45,36,43,34,41,32,39,30,37,28,47), (1,32,7,26)(2,33,8,27)(3,34,9,28)(4,35,10,29)(5,36,11,30)(6,25,12,31)(13,45,19,39)(14,46,20,40)(15,47,21,41)(16,48,22,42)(17,37,23,43)(18,38,24,44)>;
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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,48)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29), (1,21,3,23,5,13,7,15,9,17,11,19)(2,16,4,18,6,20,8,22,10,24,12,14)(25,38,35,48,33,46,31,44,29,42,27,40)(26,45,36,43,34,41,32,39,30,37,28,47), (1,32,7,26)(2,33,8,27)(3,34,9,28)(4,35,10,29)(5,36,11,30)(6,25,12,31)(13,45,19,39)(14,46,20,40)(15,47,21,41)(16,48,22,42)(17,37,23,43)(18,38,24,44) );
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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,48),(13,28),(14,27),(15,26),(16,25),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29)], [(1,21,3,23,5,13,7,15,9,17,11,19),(2,16,4,18,6,20,8,22,10,24,12,14),(25,38,35,48,33,46,31,44,29,42,27,40),(26,45,36,43,34,41,32,39,30,37,28,47)], [(1,32,7,26),(2,33,8,27),(3,34,9,28),(4,35,10,29),(5,36,11,30),(6,25,12,31),(13,45,19,39),(14,46,20,40),(15,47,21,41),(16,48,22,42),(17,37,23,43),(18,38,24,44)])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | ··· | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 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 | 6 | 6 | 6 | 6 | 18 | 2 | 2 | 4 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 12 | 4 | ··· | 4 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | 2- 1+4 | S32 | C2×S32 | Q8.15D6 | Q8○D12 | D12.25D6 |
| kernel | D12.25D6 | S3×Dic6 | D12⋊5S3 | D12⋊S3 | D6.D6 | C3×S3×Q8 | C3×Q8⋊3S3 | Q8×C3⋊S3 | S3×Q8 | Q8⋊3S3 | Dic6 | C4×S3 | D12 | C3×Q8 | C32 | Q8 | C4 | C3 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 3 | 2 | 1 | 1 | 3 | 2 | 2 | 1 |
Matrix representation of D12.25D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,8,0,0,0,5,5,0,0,0,0,8,0,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,9,11,0,0,0,0,2,4,0,0],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,3,6,0,0,0,0,7,10,0,0] >;
D12.25D6 in GAP, Magma, Sage, TeX
D_{12}._{25}D_6 % in TeX
G:=Group("D12.25D6"); // GroupNames label
G:=SmallGroup(288,963);
// by ID
G=gap.SmallGroup(288,963);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d^-1=a^6*c^5>;
// generators/relations