metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊21D4, C6.1172+ (1+4), C4⋊C4⋊10D6, (C2×Q8)⋊19D6, C22⋊Q8⋊7S3, C3⋊7(D4⋊5D4), D6.20(C2×D4), C4.111(S3×D4), D6⋊D4⋊16C2, C12⋊D4⋊25C2, (C6×Q8)⋊7C22, D6⋊C4⋊20C22, C12.234(C2×D4), C22⋊C4.57D6, Dic3⋊5D4⋊25C2, C6.76(C22×D4), D6.D4⋊17C2, C2.34(D4○D12), (C22×D12)⋊16C2, (C2×D12)⋊25C22, (C2×C12).54C23, (C2×C6).174C24, (C22×C4).252D6, C12.23D4⋊12C2, Dic3⋊C4⋊53C22, C22⋊3(Q8⋊3S3), (C4×Dic3)⋊28C22, (S3×C23).52C22, (C22×C6).202C23, C23.199(C22×S3), C22.195(S3×C23), (C22×S3).196C23, (C22×C12).254C22, (C2×Dic3).233C23, C6.D4.115C22, C2.49(C2×S3×D4), (C2×C6)⋊7(C4○D4), (C4×C3⋊D4)⋊22C2, (S3×C22⋊C4)⋊8C2, (S3×C2×C4)⋊18C22, (C3×C4⋊C4)⋊19C22, (C2×Q8⋊3S3)⋊7C2, C6.114(C2×C4○D4), (C3×C22⋊Q8)⋊10C2, (C2×C4).47(C22×S3), C2.17(C2×Q8⋊3S3), (C2×C3⋊D4).122C22, (C3×C22⋊C4).29C22, SmallGroup(192,1189)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1040 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], S3 [×7], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×18], Q8 [×2], C23, C23 [×15], Dic3 [×3], C12 [×2], C12 [×5], D6 [×4], D6 [×21], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4, C22×C4, C22×C4 [×5], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], C4×S3 [×8], D12 [×4], D12 [×12], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×4], C22×S3 [×10], C22×C6, 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, C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4 [×8], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×4], C2×D12 [×2], C2×D12 [×6], C2×D12 [×4], Q8⋊3S3 [×4], C2×C3⋊D4, C22×C12, C6×Q8, S3×C23 [×2], D4⋊5D4, S3×C22⋊C4 [×2], D6⋊D4 [×2], Dic3⋊5D4, D6.D4 [×2], C12⋊D4, C12⋊D4 [×2], C4×C3⋊D4, C12.23D4, C3×C22⋊Q8, C22×D12, C2×Q8⋊3S3, D12⋊21D4
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], Q8⋊3S3 [×2], S3×C23, D4⋊5D4, C2×S3×D4, C2×Q8⋊3S3, D4○D12, D12⋊21D4
Generators and relations
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a5, cbc-1=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 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)(26 36)(27 35)(28 34)(29 33)(30 32)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 43 22 33)(2 48 23 26)(3 41 24 31)(4 46 13 36)(5 39 14 29)(6 44 15 34)(7 37 16 27)(8 42 17 32)(9 47 18 25)(10 40 19 30)(11 45 20 35)(12 38 21 28)
(1 27)(2 32)(3 25)(4 30)(5 35)(6 28)(7 33)(8 26)(9 31)(10 36)(11 29)(12 34)(13 40)(14 45)(15 38)(16 43)(17 48)(18 41)(19 46)(20 39)(21 44)(22 37)(23 42)(24 47)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,45)(38,44)(39,43)(40,42)(46,48), (1,43,22,33)(2,48,23,26)(3,41,24,31)(4,46,13,36)(5,39,14,29)(6,44,15,34)(7,37,16,27)(8,42,17,32)(9,47,18,25)(10,40,19,30)(11,45,20,35)(12,38,21,28), (1,27)(2,32)(3,25)(4,30)(5,35)(6,28)(7,33)(8,26)(9,31)(10,36)(11,29)(12,34)(13,40)(14,45)(15,38)(16,43)(17,48)(18,41)(19,46)(20,39)(21,44)(22,37)(23,42)(24,47)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,45)(38,44)(39,43)(40,42)(46,48), (1,43,22,33)(2,48,23,26)(3,41,24,31)(4,46,13,36)(5,39,14,29)(6,44,15,34)(7,37,16,27)(8,42,17,32)(9,47,18,25)(10,40,19,30)(11,45,20,35)(12,38,21,28), (1,27)(2,32)(3,25)(4,30)(5,35)(6,28)(7,33)(8,26)(9,31)(10,36)(11,29)(12,34)(13,40)(14,45)(15,38)(16,43)(17,48)(18,41)(19,46)(20,39)(21,44)(22,37)(23,42)(24,47) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,21),(14,20),(15,19),(16,18),(22,24),(26,36),(27,35),(28,34),(29,33),(30,32),(37,45),(38,44),(39,43),(40,42),(46,48)], [(1,43,22,33),(2,48,23,26),(3,41,24,31),(4,46,13,36),(5,39,14,29),(6,44,15,34),(7,37,16,27),(8,42,17,32),(9,47,18,25),(10,40,19,30),(11,45,20,35),(12,38,21,28)], [(1,27),(2,32),(3,25),(4,30),(5,35),(6,28),(7,33),(8,26),(9,31),(10,36),(11,29),(12,34),(13,40),(14,45),(15,38),(16,43),(17,48),(18,41),(19,46),(20,39),(21,44),(22,37),(23,42),(24,47)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | Q8⋊3S3 | D4○D12 |
| kernel | D12⋊21D4 | S3×C22⋊C4 | D6⋊D4 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4×C3⋊D4 | C12.23D4 | C3×C22⋊Q8 | C22×D12 | C2×Q8⋊3S3 | C22⋊Q8 | D12 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_{21}D_4 % in TeX
G:=Group("D12:21D4"); // GroupNames label
G:=SmallGroup(192,1189);
// by ID
G=gap.SmallGroup(192,1189);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations