direct product, metacyclic, supersoluble, monomial
Aliases: C3×C4⋊Dic3, C12⋊1C12, C12⋊3Dic3, C6.22D12, C6.9Dic6, C62.18C22, C4⋊(C3×Dic3), (C3×C12)⋊4C4, C6.4(C3×D4), (C3×C6).5Q8, C6.2(C3×Q8), C32⋊7(C4⋊C4), (C6×C12).8C2, C6.8(C2×C12), (C2×C12).4C6, C2.1(C3×D12), (C3×C6).20D4, (C2×C6).43D6, (C2×C12).19S3, C22.5(S3×C6), C2.2(C3×Dic6), C2.4(C6×Dic3), (C6×Dic3).2C2, (C2×Dic3).2C6, C6.20(C2×Dic3), C3⋊2(C3×C4⋊C4), (C2×C6).8(C2×C6), (C2×C4).3(C3×S3), (C3×C6).29(C2×C4), SmallGroup(144,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊Dic3
G = < a,b,c,d | a3=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 104 in 60 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C6×Dic3, C6×C12, C3×C4⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3, S3×C6, C4⋊Dic3, C3×C4⋊C4, C3×Dic6, C3×D12, C6×Dic3, C3×C4⋊Dic3
►On 48 points
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 30 17 24)(2 25 18 19)(3 26 13 20)(4 27 14 21)(5 28 15 22)(6 29 16 23)(7 36 48 40)(8 31 43 41)(9 32 44 42)(10 33 45 37)(11 34 46 38)(12 35 47 39)
(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 33 4 36)(2 32 5 35)(3 31 6 34)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)
G:=sub<Sym(48)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (1,30,17,24)(2,25,18,19)(3,26,13,20)(4,27,14,21)(5,28,15,22)(6,29,16,23)(7,36,48,40)(8,31,43,41)(9,32,44,42)(10,33,45,37)(11,34,46,38)(12,35,47,39), (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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (1,30,17,24)(2,25,18,19)(3,26,13,20)(4,27,14,21)(5,28,15,22)(6,29,16,23)(7,36,48,40)(8,31,43,41)(9,32,44,42)(10,33,45,37)(11,34,46,38)(12,35,47,39), (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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48)], [(1,30,17,24),(2,25,18,19),(3,26,13,20),(4,27,14,21),(5,28,15,22),(6,29,16,23),(7,36,48,40),(8,31,43,41),(9,32,44,42),(10,33,45,37),(11,34,46,38),(12,35,47,39)], [(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)], [(1,33,4,36),(2,32,5,35),(3,31,6,34),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45)]])
C3×C4⋊Dic3 is a maximal subgroup of
D12⋊3Dic3 C6.17D24 Dic6⋊Dic3 C12.73D12 C12.Dic6 C12.6Dic6 C6.18D24 C12.8Dic6 C62.11C23 C62.13C23 Dic3⋊6Dic6 C62.16C23 C62.17C23 C62.18C23 C62.19C23 C62.24C23 D6⋊7Dic6 C62.28C23 C62.39C23 C12.30D12 C62.42C23 D6.D12 Dic3⋊5D12 C62.65C23 D12⋊Dic3 D6⋊4Dic6 C62.70C23 C12⋊7D12 C12⋊2D12 C12⋊3Dic6 C12⋊Dic6 C12×Dic6 C12×D12 C3×S3×C4⋊C4 C3×D4×Dic3 C3×Q8×Dic3 C62.20D6 C36⋊C12 C62.30D6
C3×C4⋊Dic3 is a maximal quotient of
C62.20D6 C36⋊C12
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12P | 12Q | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | Dic6 | D12 | C3×D4 | C3×Q8 | C3×Dic3 | S3×C6 | C3×Dic6 | C3×D12 |
| kernel | C3×C4⋊Dic3 | C6×Dic3 | C6×C12 | C4⋊Dic3 | C3×C12 | C2×Dic3 | C2×C12 | C12 | C2×C12 | C3×C6 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C3×C4⋊Dic3 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 10 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[10,0,0,0,0,4,0,0,0,0,10,0,0,0,0,4],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0] >;
C3×C4⋊Dic3 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3xC4:Dic3"); // GroupNames label
G:=SmallGroup(144,78);
// by ID
G=gap.SmallGroup(144,78);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,313,151,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations