metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊10D4, C42⋊21D6, C6.1252+ (1+4), (C2×Q8)⋊22D6, C4.70(S3×D4), (C4×D12)⋊43C2, C3⋊9(D4⋊5D4), C22⋊C4⋊33D6, D6.24(C2×D4), C12.63(C2×D4), D6⋊11(C4○D4), D6⋊3D4⋊33C2, Dic3⋊D4⋊40C2, D6⋊D4⋊24C2, (C4×C12)⋊23C22, D6⋊C4⋊54C22, D6⋊3Q8⋊28C2, (C2×D4).173D6, C4.4D4⋊10S3, (C6×Q8)⋊13C22, C6.90(C22×D4), C23.9D6⋊42C2, C2.49(D4○D12), (C2×D12)⋊28C22, (C2×C6).220C24, C4⋊Dic3⋊60C22, (C2×C12).601C23, Dic3⋊C4⋊26C22, (C6×D4).155C22, (C22×C6).50C23, C23.52(C22×S3), C6.D4⋊33C22, (S3×C23).64C22, C22.241(S3×C23), (C22×S3).215C23, (C2×Dic3).115C23, (C2×S3×D4)⋊17C2, C2.63(C2×S3×D4), (S3×C2×C4)⋊26C22, C2.76(S3×C4○D4), (S3×C22⋊C4)⋊17C2, C6.187(C2×C4○D4), (C2×Q8⋊3S3)⋊11C2, (C3×C4.4D4)⋊12C2, (C2×C3⋊D4)⋊23C22, (C3×C22⋊C4)⋊29C22, (C2×C4).195(C22×S3), SmallGroup(192,1235)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1024 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×29], S3 [×7], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], D6 [×17], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×6], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], C4×S3 [×10], D12 [×4], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×3], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×S3 [×2], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×4], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×2], S3×C2×C4 [×4], C2×D12 [×2], C2×D12 [×2], S3×D4 [×4], Q8⋊3S3 [×4], C2×C3⋊D4 [×4], C6×D4, C6×Q8, S3×C23 [×2], D4⋊5D4, C4×D12 [×2], S3×C22⋊C4 [×2], D6⋊D4 [×2], C23.9D6 [×2], Dic3⋊D4 [×2], D6⋊3D4, D6⋊3Q8, C3×C4.4D4, C2×S3×D4, C2×Q8⋊3S3, D12⋊10D4
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), S3×D4 [×2], S3×C23, D4⋊5D4, C2×S3×D4, S3×C4○D4, D4○D12, D12⋊10D4
Generators and relations
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=a6b, dbd=a10b, 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 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)
(1 34 41 16)(2 35 42 17)(3 36 43 18)(4 25 44 19)(5 26 45 20)(6 27 46 21)(7 28 47 22)(8 29 48 23)(9 30 37 24)(10 31 38 13)(11 32 39 14)(12 33 40 15)
(1 28)(2 33)(3 26)(4 31)(5 36)(6 29)(7 34)(8 27)(9 32)(10 25)(11 30)(12 35)(13 44)(14 37)(15 42)(16 47)(17 40)(18 45)(19 38)(20 43)(21 48)(22 41)(23 46)(24 39)
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)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47), (1,34,41,16)(2,35,42,17)(3,36,43,18)(4,25,44,19)(5,26,45,20)(6,27,46,21)(7,28,47,22)(8,29,48,23)(9,30,37,24)(10,31,38,13)(11,32,39,14)(12,33,40,15), (1,28)(2,33)(3,26)(4,31)(5,36)(6,29)(7,34)(8,27)(9,32)(10,25)(11,30)(12,35)(13,44)(14,37)(15,42)(16,47)(17,40)(18,45)(19,38)(20,43)(21,48)(22,41)(23,46)(24,39)>;
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)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47), (1,34,41,16)(2,35,42,17)(3,36,43,18)(4,25,44,19)(5,26,45,20)(6,27,46,21)(7,28,47,22)(8,29,48,23)(9,30,37,24)(10,31,38,13)(11,32,39,14)(12,33,40,15), (1,28)(2,33)(3,26)(4,31)(5,36)(6,29)(7,34)(8,27)(9,32)(10,25)(11,30)(12,35)(13,44)(14,37)(15,42)(16,47)(17,40)(18,45)(19,38)(20,43)(21,48)(22,41)(23,46)(24,39) );
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),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47)], [(1,34,41,16),(2,35,42,17),(3,36,43,18),(4,25,44,19),(5,26,45,20),(6,27,46,21),(7,28,47,22),(8,29,48,23),(9,30,37,24),(10,31,38,13),(11,32,39,14),(12,33,40,15)], [(1,28),(2,33),(3,26),(4,31),(5,36),(6,29),(7,34),(8,27),(9,32),(10,25),(11,30),(12,35),(13,44),(14,37),(15,42),(16,47),(17,40),(18,45),(19,38),(20,43),(21,48),(22,41),(23,46),(24,39)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 8 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,3,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,10,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | ··· | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | S3×D4 | S3×C4○D4 | D4○D12 |
| kernel | D12⋊10D4 | C4×D12 | S3×C22⋊C4 | D6⋊D4 | C23.9D6 | Dic3⋊D4 | D6⋊3D4 | D6⋊3Q8 | C3×C4.4D4 | C2×S3×D4 | C2×Q8⋊3S3 | C4.4D4 | D12 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_{10}D_4 % in TeX
G:=Group("D12:10D4"); // GroupNames label
G:=SmallGroup(192,1235);
// by ID
G=gap.SmallGroup(192,1235);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,192,6278]);
// 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,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations