direct product, metabelian, nilpotent (class 2), monomial
Aliases: D4×C3×C6, C4⋊C62, C22⋊2C62, C62⋊6C22, C12⋊4(C2×C6), (C2×C12)⋊6C6, (C6×C12)⋊10C2, (C2×C62)⋊1C2, C23⋊2(C3×C6), (C22×C6)⋊3C6, C2.1(C2×C62), (C3×C12)⋊10C22, C6.14(C22×C6), (C3×C6).39C23, (C2×C4)⋊2(C3×C6), (C2×C6)⋊4(C2×C6), SmallGroup(144,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C3×C6
G = < a,b,c,d | a3=b6=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 210 in 162 conjugacy classes, 114 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, D4, C23, C32, C12, C2×C6, C2×C6, C2×D4, C3×C6, C3×C6, C3×C6, C2×C12, C3×D4, C22×C6, C3×C12, C62, C62, C62, C6×D4, C6×C12, D4×C32, C2×C62, D4×C3×C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, C62, C6×D4, D4×C32, C2×C62, D4×C3×C6
►On 72 points
(1 40 35)(2 41 36)(3 42 31)(4 37 32)(5 38 33)(6 39 34)(7 17 28)(8 18 29)(9 13 30)(10 14 25)(11 15 26)(12 16 27)(19 71 50)(20 72 51)(21 67 52)(22 68 53)(23 69 54)(24 70 49)(43 56 64)(44 57 65)(45 58 66)(46 59 61)(47 60 62)(48 55 63)
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)
(1 58 10 50)(2 59 11 51)(3 60 12 52)(4 55 7 53)(5 56 8 54)(6 57 9 49)(13 24 39 65)(14 19 40 66)(15 20 41 61)(16 21 42 62)(17 22 37 63)(18 23 38 64)(25 71 35 45)(26 72 36 46)(27 67 31 47)(28 68 32 48)(29 69 33 43)(30 70 34 44)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 50)(8 51)(9 52)(10 53)(11 54)(12 49)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(25 68)(26 69)(27 70)(28 71)(29 72)(30 67)(31 44)(32 45)(33 46)(34 47)(35 48)(36 43)(37 66)(38 61)(39 62)(40 63)(41 64)(42 65)
G:=sub<Sym(72)| (1,40,35)(2,41,36)(3,42,31)(4,37,32)(5,38,33)(6,39,34)(7,17,28)(8,18,29)(9,13,30)(10,14,25)(11,15,26)(12,16,27)(19,71,50)(20,72,51)(21,67,52)(22,68,53)(23,69,54)(24,70,49)(43,56,64)(44,57,65)(45,58,66)(46,59,61)(47,60,62)(48,55,63), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,58,10,50)(2,59,11,51)(3,60,12,52)(4,55,7,53)(5,56,8,54)(6,57,9,49)(13,24,39,65)(14,19,40,66)(15,20,41,61)(16,21,42,62)(17,22,37,63)(18,23,38,64)(25,71,35,45)(26,72,36,46)(27,67,31,47)(28,68,32,48)(29,69,33,43)(30,70,34,44), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,50)(8,51)(9,52)(10,53)(11,54)(12,49)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,44)(32,45)(33,46)(34,47)(35,48)(36,43)(37,66)(38,61)(39,62)(40,63)(41,64)(42,65)>;
G:=Group( (1,40,35)(2,41,36)(3,42,31)(4,37,32)(5,38,33)(6,39,34)(7,17,28)(8,18,29)(9,13,30)(10,14,25)(11,15,26)(12,16,27)(19,71,50)(20,72,51)(21,67,52)(22,68,53)(23,69,54)(24,70,49)(43,56,64)(44,57,65)(45,58,66)(46,59,61)(47,60,62)(48,55,63), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,58,10,50)(2,59,11,51)(3,60,12,52)(4,55,7,53)(5,56,8,54)(6,57,9,49)(13,24,39,65)(14,19,40,66)(15,20,41,61)(16,21,42,62)(17,22,37,63)(18,23,38,64)(25,71,35,45)(26,72,36,46)(27,67,31,47)(28,68,32,48)(29,69,33,43)(30,70,34,44), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,50)(8,51)(9,52)(10,53)(11,54)(12,49)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,44)(32,45)(33,46)(34,47)(35,48)(36,43)(37,66)(38,61)(39,62)(40,63)(41,64)(42,65) );
G=PermutationGroup([[(1,40,35),(2,41,36),(3,42,31),(4,37,32),(5,38,33),(6,39,34),(7,17,28),(8,18,29),(9,13,30),(10,14,25),(11,15,26),(12,16,27),(19,71,50),(20,72,51),(21,67,52),(22,68,53),(23,69,54),(24,70,49),(43,56,64),(44,57,65),(45,58,66),(46,59,61),(47,60,62),(48,55,63)], [(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,58,10,50),(2,59,11,51),(3,60,12,52),(4,55,7,53),(5,56,8,54),(6,57,9,49),(13,24,39,65),(14,19,40,66),(15,20,41,61),(16,21,42,62),(17,22,37,63),(18,23,38,64),(25,71,35,45),(26,72,36,46),(27,67,31,47),(28,68,32,48),(29,69,33,43),(30,70,34,44)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,50),(8,51),(9,52),(10,53),(11,54),(12,49),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20),(25,68),(26,69),(27,70),(28,71),(29,72),(30,67),(31,44),(32,45),(33,46),(34,47),(35,48),(36,43),(37,66),(38,61),(39,62),(40,63),(41,64),(42,65)]])
D4×C3×C6 is a maximal subgroup of
C62.116D4 (C6×D4).S3 C62.38D4 C62.131D4 C62.72D4 C62.254C23 C62⋊13D4 C62.256C23 C62⋊14D4 C62.258C23 C32⋊82+ 1+4
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6X | 6Y | ··· | 6BD | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 |
| kernel | D4×C3×C6 | C6×C12 | D4×C32 | C2×C62 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C3×C6 | C6 |
| # reps | 1 | 1 | 4 | 2 | 8 | 8 | 32 | 16 | 2 | 16 |
Matrix representation of D4×C3×C6 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 12 | 0 | 0 |
| 0 | 12 | 11 |
| 0 | 1 | 1 |
| 12 | 0 | 0 |
| 0 | 12 | 11 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(13))| [3,0,0,0,9,0,0,0,9],[12,0,0,0,4,0,0,0,4],[12,0,0,0,12,1,0,11,1],[12,0,0,0,12,0,0,11,1] >;
D4×C3×C6 in GAP, Magma, Sage, TeX
D_4\times C_3\times C_6
% in TeX
G:=Group("D4xC3xC6"); // GroupNames label
G:=SmallGroup(144,179);
// by ID
G=gap.SmallGroup(144,179);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-2,889]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations