metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊23D4, C42⋊16D6, C6.1032+ (1+4), C4⋊C4⋊47D6, (C4×D4)⋊12S3, Dic3⋊D4⋊7C2, (D4×C12)⋊14C2, (C4×D12)⋊28C2, C3⋊2(D4⋊5D4), C22⋊C4⋊46D6, D6.14(C2×D4), C4.139(S3×D4), (C22×C4)⋊15D6, C23⋊2D6⋊20C2, C12⋊7D4⋊18C2, (C4×C12)⋊19C22, D6⋊C4⋊30C22, D6.D4⋊7C2, (C2×D4).213D6, C12.345(C2×D4), (C22×D12)⋊9C2, (C2×C6).94C24, C6.49(C22×D4), C42⋊7S3⋊16C2, C2.15(D4○D12), C22⋊4(C4○D12), Dic3⋊C4⋊3C22, C4⋊Dic3⋊59C22, C12.48D4⋊10C2, (C2×C12).782C23, (C22×C12)⋊16C22, (C2×Dic6)⋊53C22, (C6×D4).305C22, (C2×D12).210C22, (S3×C23).39C22, (C22×C6).164C23, C22.119(S3×C23), C23.104(C22×S3), (C2×Dic3).40C23, (C22×S3).172C23, C6.D4.11C22, C2.22(C2×S3×D4), (C2×C4○D12)⋊7C2, (C2×C6)⋊2(C4○D4), (S3×C2×C4)⋊48C22, C6.41(C2×C4○D4), (C3×C4⋊C4)⋊59C22, (S3×C22⋊C4)⋊28C2, C2.45(C2×C4○D12), (C2×C3⋊D4)⋊3C22, (C3×C22⋊C4)⋊56C22, (C2×C4).158(C22×S3), SmallGroup(192,1109)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1016 in 334 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], S3 [×6], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×4], C12 [×2], C12 [×4], D6 [×4], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×6], D12 [×4], D12 [×6], C2×Dic3 [×4], C3⋊D4 [×6], C2×C12 [×5], C2×C12 [×4], C3×D4 [×2], C22×S3 [×4], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×8], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×4], C2×D12 [×2], C2×D12 [×2], C2×D12 [×4], C4○D12 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, S3×C23 [×2], D4⋊5D4, C4×D12, C42⋊7S3, S3×C22⋊C4 [×2], Dic3⋊D4 [×2], D6.D4 [×2], C12.48D4, C12⋊7D4, C23⋊2D6 [×2], D4×C12, C22×D12, C2×C4○D12, D12⋊23D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C4○D12 [×2], S3×D4 [×2], S3×C23, D4⋊5D4, C2×C4○D12, C2×S3×D4, D4○D12, D12⋊23D4
Generators and relations
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a6b, bd=db, dcd=c-1 >
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)
(1 17 25 47)(2 18 26 48)(3 19 27 37)(4 20 28 38)(5 21 29 39)(6 22 30 40)(7 23 31 41)(8 24 32 42)(9 13 33 43)(10 14 34 44)(11 15 35 45)(12 16 36 46)
(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44), (1,17,25,47)(2,18,26,48)(3,19,27,37)(4,20,28,38)(5,21,29,39)(6,22,30,40)(7,23,31,41)(8,24,32,42)(9,13,33,43)(10,14,34,44)(11,15,35,45)(12,16,36,46), (13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44), (1,17,25,47)(2,18,26,48)(3,19,27,37)(4,20,28,38)(5,21,29,39)(6,22,30,40)(7,23,31,41)(8,24,32,42)(9,13,33,43)(10,14,34,44)(11,15,35,45)(12,16,36,46), (13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44)], [(1,17,25,47),(2,18,26,48),(3,19,27,37),(4,20,28,38),(5,21,29,39),(6,22,30,40),(7,23,31,41),(8,24,32,42),(9,13,33,43),(10,14,34,44),(11,15,35,45),(12,16,36,46)], [(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 7 | 10 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,3,3,0,0,10,6],[12,0,0,0,0,12,0,0,0,0,3,7,0,0,10,10],[0,1,0,0,12,0,0,0,0,0,2,9,0,0,4,11],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | S3×D4 | D4○D12 |
| kernel | D12⋊23D4 | C4×D12 | C42⋊7S3 | S3×C22⋊C4 | Dic3⋊D4 | D6.D4 | C12.48D4 | C12⋊7D4 | C23⋊2D6 | D4×C12 | C22×D12 | C2×C4○D12 | C4×D4 | D12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C22 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_{23}D_4 % in TeX
G:=Group("D12:23D4"); // GroupNames label
G:=SmallGroup(192,1109);
// by ID
G=gap.SmallGroup(192,1109);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations