direct product, metabelian, supersoluble, monomial
Aliases: C22×C3⋊D12, C62⋊16D4, C62.146C23, C6⋊3(C2×D12), (C2×C6)⋊12D12, C23.47S32, (S3×C23)⋊7S3, (S3×C6)⋊7C23, D6⋊6(C22×S3), C3⋊3(C22×D12), C32⋊6(C22×D4), (C2×Dic3)⋊22D6, (C22×S3)⋊15D6, (C3×C6).33C24, C6.33(S3×C23), Dic3⋊5(C22×S3), (C3×Dic3)⋊5C23, (C22×C6).123D6, (C6×Dic3)⋊27C22, (C22×Dic3)⋊12S3, (C2×C62).81C22, (C3×C6)⋊5(C2×D4), C6⋊1(C2×C3⋊D4), (S3×C22×C6)⋊6C2, (C23×C3⋊S3)⋊4C2, (C2×C3⋊S3)⋊4C23, (S3×C2×C6)⋊17C22, C22.70(C2×S32), C2.33(C22×S32), (Dic3×C2×C6)⋊12C2, C3⋊1(C22×C3⋊D4), (C2×C6)⋊12(C3⋊D4), (C22×C3⋊S3)⋊13C22, (C2×C6).161(C22×S3), SmallGroup(288,974)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C3⋊D12
G = < a,b,c,d,e | a2=b2=c3=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 2178 in 539 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×D4, C3×Dic3, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, S3×C23, S3×C23, C23×C6, C3⋊D12, C6×Dic3, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C22×D12, C22×C3⋊D4, C2×C3⋊D12, Dic3×C2×C6, S3×C22×C6, C23×C3⋊S3, C22×C3⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C3⋊D4, C22×S3, C22×D4, S32, C2×D12, C2×C3⋊D4, S3×C23, C3⋊D12, C2×S32, C22×D12, C22×C3⋊D4, C2×C3⋊D12, C22×S32, C22×C3⋊D12
►On 48 points
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 44)(26 45)(27 46)(28 47)(29 48)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 17 21)(14 22 18)(15 19 23)(16 24 20)(25 33 29)(26 30 34)(27 35 31)(28 32 36)(37 41 45)(38 46 42)(39 43 47)(40 48 44)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)
G:=sub<Sym(48)| (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,44)(26,45)(27,46)(28,47)(29,48)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20)(25,33,29)(26,30,34)(27,35,31)(28,32,36)(37,41,45)(38,46,42)(39,43,47)(40,48,44), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33)>;
G:=Group( (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,44)(26,45)(27,46)(28,47)(29,48)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20)(25,33,29)(26,30,34)(27,35,31)(28,32,36)(37,41,45)(38,46,42)(39,43,47)(40,48,44), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,44),(26,45),(27,46),(28,47),(29,48),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(36,43)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,25),(12,26),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,9,5),(2,6,10),(3,11,7),(4,8,12),(13,17,21),(14,22,18),(15,19,23),(16,24,20),(25,33,29),(26,30,34),(27,35,31),(28,32,36),(37,41,45),(38,46,42),(39,43,47),(40,48,44)], [(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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,36),(22,35),(23,34),(24,33)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | ··· | 6N | 6O | ··· | 6U | 6V | ··· | 6AC | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | D12 | C3⋊D4 | S32 | C3⋊D12 | C2×S32 |
| kernel | C22×C3⋊D12 | C2×C3⋊D12 | Dic3×C2×C6 | S3×C22×C6 | C23×C3⋊S3 | C22×Dic3 | S3×C23 | C62 | C2×Dic3 | C22×S3 | C22×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 8 | 8 | 1 | 4 | 3 |
Matrix representation of C22×C3⋊D12 ►in GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1] >;
C22×C3⋊D12 in GAP, Magma, Sage, TeX
C_2^2\times C_3\rtimes D_{12} % in TeX
G:=Group("C2^2xC3:D12"); // GroupNames label
G:=SmallGroup(288,974);
// by ID
G=gap.SmallGroup(288,974);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=e^2=1,a*b=b*a,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=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations