metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.36D4, C4⋊C4⋊4D6, (C2×Q8)⋊4D6, (C2×C6)⋊5SD16, C22⋊Q8⋊1S3, (C2×C12).77D4, C4.101(S3×D4), (C6×Q8)⋊1C22, C6.47C22≀C2, C12.152(C2×D4), C6.D8⋊37C2, C3⋊3(C22⋊SD16), C6.72(C2×SD16), (C22×C6).92D4, (C22×C4).143D6, C2.15(C23⋊2D6), C2.15(D4⋊D6), C12.55D4⋊13C2, C6.116(C8⋊C22), (C2×C12).365C23, C22⋊3(Q8⋊2S3), (C22×D12).13C2, C23.69(C3⋊D4), (C2×D12).242C22, (C22×C12).169C22, (C2×C3⋊C8)⋊6C22, (C3×C4⋊C4)⋊6C22, (C3×C22⋊Q8)⋊1C2, (C2×Q8⋊2S3)⋊8C2, (C2×C6).496(C2×D4), C2.9(C2×Q8⋊2S3), (C2×C4).55(C3⋊D4), (C2×C4).465(C22×S3), C22.171(C2×C3⋊D4), SmallGroup(192,605)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.36D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd=a6c-1 >
Subgroups: 704 in 188 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], S3 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×10], C12 [×2], C12 [×3], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×7], C2×Q8, C24, C3⋊C8 [×2], D12 [×4], D12 [×6], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×10], C22×C6, C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C2×C3⋊C8 [×2], Q8⋊2S3 [×4], C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×D12 [×2], C2×D12 [×5], C22×C12, C6×Q8, S3×C23, C22⋊SD16, C6.D8 [×2], C12.55D4, C2×Q8⋊2S3 [×2], C3×C22⋊Q8, C22×D12, D12.36D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×2], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C2×SD16, C8⋊C22, Q8⋊2S3 [×2], S3×D4 [×2], C2×C3⋊D4, C22⋊SD16, C23⋊2D6, C2×Q8⋊2S3, D4⋊D6, D12.36D4
►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 23)(14 22)(15 21)(16 20)(17 19)(25 33)(26 32)(27 31)(28 30)(34 36)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)
(1 31 38 20)(2 26 39 15)(3 33 40 22)(4 28 41 17)(5 35 42 24)(6 30 43 19)(7 25 44 14)(8 32 45 21)(9 27 46 16)(10 34 47 23)(11 29 48 18)(12 36 37 13)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 36)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
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)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44), (1,31,38,20)(2,26,39,15)(3,33,40,22)(4,28,41,17)(5,35,42,24)(6,30,43,19)(7,25,44,14)(8,32,45,21)(9,27,46,16)(10,34,47,23)(11,29,48,18)(12,36,37,13), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,36)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)>;
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)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44), (1,31,38,20)(2,26,39,15)(3,33,40,22)(4,28,41,17)(5,35,42,24)(6,30,43,19)(7,25,44,14)(8,32,45,21)(9,27,46,16)(10,34,47,23)(11,29,48,18)(12,36,37,13), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,36)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48) );
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),(13,23),(14,22),(15,21),(16,20),(17,19),(25,33),(26,32),(27,31),(28,30),(34,36),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44)], [(1,31,38,20),(2,26,39,15),(3,33,40,22),(4,28,41,17),(5,35,42,24),(6,30,43,19),(7,25,44,14),(8,32,45,21),(9,27,46,16),(10,34,47,23),(11,29,48,18),(12,36,37,13)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,36),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | SD16 | C3⋊D4 | C3⋊D4 | C8⋊C22 | S3×D4 | Q8⋊2S3 | D4⋊D6 |
| kernel | D12.36D4 | C6.D8 | C12.55D4 | C2×Q8⋊2S3 | C3×C22⋊Q8 | C22×D12 | C22⋊Q8 | D12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of D12.36D4 ►in GL6(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 43 | 13 | 0 | 0 | 0 | 0 |
| 60 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 2 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 67 | 67 |
| 0 | 0 | 0 | 0 | 67 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,67,67,0,0,0,0,67,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
D12.36D4 in GAP, Magma, Sage, TeX
D_{12}._{36}D_4 % in TeX
G:=Group("D12.36D4"); // GroupNames label
G:=SmallGroup(192,605);
// by ID
G=gap.SmallGroup(192,605);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,1123,297,136,6278]);
// 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=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^6*c^-1>;
// generators/relations