Aliases: D12.A4, SL2(𝔽3).12D6, Q8○D12⋊C3, C4.A4⋊4S3, C4.3(S3×A4), (S3×Q8)⋊2C6, C12.3(C2×A4), D6.2(C2×A4), C3⋊2(D4.A4), C6.9(C22×A4), Q8.10(S3×C6), (S3×SL2(𝔽3))⋊5C2, (C3×SL2(𝔽3)).12C22, C2.10(C2×S3×A4), (C3×C4○D4)⋊2C6, (C3×C4.A4)⋊5C2, C4○D4⋊3(C3×S3), (C3×Q8).5(C2×C6), SmallGroup(288,926)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — D12.A4 |
| C3×Q8 — D12.A4 |
Generators and relations for D12.A4
G = < a,b,c,d,e | a12=b2=e3=1, c2=d2=a6, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a6c, ece-1=a6cd, ede-1=c >
Subgroups: 446 in 97 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C4○D4, C4○D4, C3×S3, C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, 2- 1+4, C3×C12, S3×C6, C2×SL2(𝔽3), C4.A4, C4.A4, C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, C3×C4○D4, C3×SL2(𝔽3), C3×D12, D4.A4, Q8○D12, S3×SL2(𝔽3), C3×C4.A4, D12.A4
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C2×A4, S3×C6, C22×A4, S3×A4, D4.A4, C2×S3×A4, D12.A4
►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 19)(14 18)(15 17)(20 24)(21 23)(25 29)(26 28)(30 36)(31 35)(32 34)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 40 7 46)(2 41 8 47)(3 42 9 48)(4 43 10 37)(5 44 11 38)(6 45 12 39)(13 30 19 36)(14 31 20 25)(15 32 21 26)(16 33 22 27)(17 34 23 28)(18 35 24 29)
(1 26 7 32)(2 27 8 33)(3 28 9 34)(4 29 10 35)(5 30 11 36)(6 31 12 25)(13 38 19 44)(14 39 20 45)(15 40 21 46)(16 41 22 47)(17 42 23 48)(18 43 24 37)
(13 38 36)(14 39 25)(15 40 26)(16 41 27)(17 42 28)(18 43 29)(19 44 30)(20 45 31)(21 46 32)(22 47 33)(23 48 34)(24 37 35)
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)(13,19)(14,18)(15,17)(20,24)(21,23)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48), (1,40,7,46)(2,41,8,47)(3,42,9,48)(4,43,10,37)(5,44,11,38)(6,45,12,39)(13,30,19,36)(14,31,20,25)(15,32,21,26)(16,33,22,27)(17,34,23,28)(18,35,24,29), (1,26,7,32)(2,27,8,33)(3,28,9,34)(4,29,10,35)(5,30,11,36)(6,31,12,25)(13,38,19,44)(14,39,20,45)(15,40,21,46)(16,41,22,47)(17,42,23,48)(18,43,24,37), (13,38,36)(14,39,25)(15,40,26)(16,41,27)(17,42,28)(18,43,29)(19,44,30)(20,45,31)(21,46,32)(22,47,33)(23,48,34)(24,37,35)>;
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)(13,19)(14,18)(15,17)(20,24)(21,23)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48), (1,40,7,46)(2,41,8,47)(3,42,9,48)(4,43,10,37)(5,44,11,38)(6,45,12,39)(13,30,19,36)(14,31,20,25)(15,32,21,26)(16,33,22,27)(17,34,23,28)(18,35,24,29), (1,26,7,32)(2,27,8,33)(3,28,9,34)(4,29,10,35)(5,30,11,36)(6,31,12,25)(13,38,19,44)(14,39,20,45)(15,40,21,46)(16,41,22,47)(17,42,23,48)(18,43,24,37), (13,38,36)(14,39,25)(15,40,26)(16,41,27)(17,42,28)(18,43,29)(19,44,30)(20,45,31)(21,46,32)(22,47,33)(23,48,34)(24,37,35) );
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),(13,19),(14,18),(15,17),(20,24),(21,23),(25,29),(26,28),(30,36),(31,35),(32,34),(37,45),(38,44),(39,43),(40,42),(46,48)], [(1,40,7,46),(2,41,8,47),(3,42,9,48),(4,43,10,37),(5,44,11,38),(6,45,12,39),(13,30,19,36),(14,31,20,25),(15,32,21,26),(16,33,22,27),(17,34,23,28),(18,35,24,29)], [(1,26,7,32),(2,27,8,33),(3,28,9,34),(4,29,10,35),(5,30,11,36),(6,31,12,25),(13,38,19,44),(14,39,20,45),(15,40,21,46),(16,41,22,47),(17,42,23,48),(18,43,24,37)], [(13,38,36),(14,39,25),(15,40,26),(16,41,27),(17,42,28),(18,43,29),(19,44,30),(20,45,31),(21,46,32),(22,47,33),(23,48,34),(24,37,35)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | ··· | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | ··· | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 6 | 2 | 4 | 4 | 8 | 8 | 2 | 6 | 18 | 18 | 2 | 4 | 4 | 8 | 8 | 12 | 24 | 24 | 24 | 24 | 2 | 2 | 8 | ··· | 8 | 12 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | A4 | C2×A4 | C2×A4 | D4.A4 | D4.A4 | D12.A4 | D12.A4 | S3×A4 | C2×S3×A4 |
| kernel | D12.A4 | S3×SL2(𝔽3) | C3×C4.A4 | Q8○D12 | S3×Q8 | C3×C4○D4 | C4.A4 | SL2(𝔽3) | C4○D4 | Q8 | D12 | C12 | D6 | C3 | C3 | C1 | C1 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 1 |
Matrix representation of D12.A4 ►in GL4(𝔽13) generated by
| 3 | 10 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 1 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 10 | 0 | 9 | 0 |
| 0 | 10 | 0 | 9 |
| 9 | 0 | 3 | 0 |
| 0 | 9 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 9 | 0 | 3 | 0 |
| 0 | 9 | 0 | 3 |
G:=sub<GL(4,GF(13))| [3,3,0,0,10,6,0,0,0,0,3,3,0,0,10,6],[1,0,0,0,1,12,0,0,0,0,1,0,0,0,1,12],[0,0,12,0,0,0,0,12,1,0,0,0,0,1,0,0],[10,0,9,0,0,10,0,9,9,0,3,0,0,9,0,3],[1,0,9,0,0,1,0,9,0,0,3,0,0,0,0,3] >;
D12.A4 in GAP, Magma, Sage, TeX
D_{12}.A_4 % in TeX
G:=Group("D12.A4"); // GroupNames label
G:=SmallGroup(288,926);
// by ID
G=gap.SmallGroup(288,926);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-3,-2,2045,1016,269,360,123,515,242,4037]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=e^3=1,c^2=d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^6*c,e*c*e^-1=a^6*c*d,e*d*e^-1=c>;
// generators/relations