direct product, metabelian, supersoluble, monomial
Aliases: C3×D4⋊D6, C62.66D4, D4⋊S3⋊6C6, D4⋊4(S3×C6), Q8⋊6(S3×C6), (C3×D4)⋊22D6, (C3×Q8)⋊23D6, C6.59(C6×D4), (C2×D12)⋊10C6, (C6×D12)⋊12C2, Q8⋊2S3⋊6C6, C12.58(C3×D4), C4.Dic3⋊9C6, D12.11(C2×C6), (C3×C12).160D4, (C2×C12).247D6, C32⋊24(C8⋊C22), (C3×C12).88C23, C12.17(C22×C6), C12.141(C3⋊D4), (C6×C12).131C22, C12.168(C22×S3), (C3×D12).40C22, (D4×C32)⋊15C22, (Q8×C32)⋊14C22, C3⋊C8⋊4(C2×C6), C4.17(S3×C2×C6), (C3×C4○D4)⋊5C6, C4○D4⋊5(C3×S3), (C3×D4)⋊4(C2×C6), C3⋊5(C3×C8⋊C22), (C3×Q8)⋊6(C2×C6), (C2×C6).9(C3×D4), (C3×D4⋊S3)⋊14C2, (C3×C4○D4)⋊10S3, (C3×C3⋊C8)⋊21C22, (C2×C4).20(S3×C6), C2.23(C6×C3⋊D4), C4.24(C3×C3⋊D4), (C2×C12).42(C2×C6), (C3×C6).267(C2×D4), (C32×C4○D4)⋊1C2, C6.160(C2×C3⋊D4), (C3×C4.Dic3)⋊7C2, C22.5(C3×C3⋊D4), (C3×Q8⋊2S3)⋊12C2, (C2×C6).46(C3⋊D4), SmallGroup(288,720)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊D6
G = < a,b,c,d,e | a3=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >
Subgroups: 418 in 156 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C32, C12, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, D12, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×C12, C3×C12, S3×C6, C62, C62, C4.Dic3, D4⋊S3, Q8⋊2S3, C3×M4(2), C3×D8, C3×SD16, C2×D12, C6×D4, C3×C4○D4, C3×C4○D4, C3×C3⋊C8, C3×D12, C3×D12, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, S3×C2×C6, D4⋊D6, C3×C8⋊C22, C3×C4.Dic3, C3×D4⋊S3, C3×Q8⋊2S3, C6×D12, C32×C4○D4, C3×D4⋊D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C8⋊C22, S3×C6, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, D4⋊D6, C3×C8⋊C22, C6×C3⋊D4, C3×D4⋊D6
►On 48 points
(1 3 2)(4 6 5)(7 8 9)(10 12 11)(13 14 15)(16 17 18)(19 21 20)(22 23 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(1 11 5 19)(2 12 6 20)(3 10 4 21)(7 14 16 23)(8 15 17 24)(9 13 18 22)(25 47 28 44)(26 48 29 45)(27 43 30 46)(31 40 34 37)(32 41 35 38)(33 42 36 39)
(1 43)(2 47)(3 45)(4 48)(5 46)(6 44)(7 34)(8 32)(9 36)(10 29)(11 27)(12 25)(13 42)(14 40)(15 38)(16 31)(17 35)(18 33)(19 30)(20 28)(21 26)(22 39)(23 37)(24 41)
(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 9)(2 8)(3 7)(4 16)(5 18)(6 17)(10 23)(11 22)(12 24)(13 19)(14 21)(15 20)(25 35)(26 34)(27 33)(28 32)(29 31)(30 36)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)
G:=sub<Sym(48)| (1,3,2)(4,6,5)(7,8,9)(10,12,11)(13,14,15)(16,17,18)(19,21,20)(22,23,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,11,5,19)(2,12,6,20)(3,10,4,21)(7,14,16,23)(8,15,17,24)(9,13,18,22)(25,47,28,44)(26,48,29,45)(27,43,30,46)(31,40,34,37)(32,41,35,38)(33,42,36,39), (1,43)(2,47)(3,45)(4,48)(5,46)(6,44)(7,34)(8,32)(9,36)(10,29)(11,27)(12,25)(13,42)(14,40)(15,38)(16,31)(17,35)(18,33)(19,30)(20,28)(21,26)(22,39)(23,37)(24,41), (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), (1,9)(2,8)(3,7)(4,16)(5,18)(6,17)(10,23)(11,22)(12,24)(13,19)(14,21)(15,20)(25,35)(26,34)(27,33)(28,32)(29,31)(30,36)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46)>;
G:=Group( (1,3,2)(4,6,5)(7,8,9)(10,12,11)(13,14,15)(16,17,18)(19,21,20)(22,23,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,11,5,19)(2,12,6,20)(3,10,4,21)(7,14,16,23)(8,15,17,24)(9,13,18,22)(25,47,28,44)(26,48,29,45)(27,43,30,46)(31,40,34,37)(32,41,35,38)(33,42,36,39), (1,43)(2,47)(3,45)(4,48)(5,46)(6,44)(7,34)(8,32)(9,36)(10,29)(11,27)(12,25)(13,42)(14,40)(15,38)(16,31)(17,35)(18,33)(19,30)(20,28)(21,26)(22,39)(23,37)(24,41), (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), (1,9)(2,8)(3,7)(4,16)(5,18)(6,17)(10,23)(11,22)(12,24)(13,19)(14,21)(15,20)(25,35)(26,34)(27,33)(28,32)(29,31)(30,36)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46) );
G=PermutationGroup([[(1,3,2),(4,6,5),(7,8,9),(10,12,11),(13,14,15),(16,17,18),(19,21,20),(22,23,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,45,47),(44,46,48)], [(1,11,5,19),(2,12,6,20),(3,10,4,21),(7,14,16,23),(8,15,17,24),(9,13,18,22),(25,47,28,44),(26,48,29,45),(27,43,30,46),(31,40,34,37),(32,41,35,38),(33,42,36,39)], [(1,43),(2,47),(3,45),(4,48),(5,46),(6,44),(7,34),(8,32),(9,36),(10,29),(11,27),(12,25),(13,42),(14,40),(15,38),(16,31),(17,35),(18,33),(19,30),(20,28),(21,26),(22,39),(23,37),(24,41)], [(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)], [(1,9),(2,8),(3,7),(4,16),(5,18),(6,17),(10,23),(11,22),(12,24),(13,19),(14,21),(15,20),(25,35),(26,34),(27,33),(28,32),(29,31),(30,36),(37,45),(38,44),(39,43),(40,48),(41,47),(42,46)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6G | 6H | ··· | 6R | 6S | 6T | 6U | 6V | 8A | 8B | 12A | ··· | 12J | 12K | ··· | 12U | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | S3×C6 | C3×C3⋊D4 | C3×C3⋊D4 | C8⋊C22 | D4⋊D6 | C3×C8⋊C22 | C3×D4⋊D6 |
| kernel | C3×D4⋊D6 | C3×C4.Dic3 | C3×D4⋊S3 | C3×Q8⋊2S3 | C6×D12 | C32×C4○D4 | D4⋊D6 | C4.Dic3 | D4⋊S3 | Q8⋊2S3 | C2×D12 | C3×C4○D4 | C3×C4○D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×D4⋊D6 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 27 | 0 | 0 | 0 |
| 38 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 35 | 27 |
| 35 | 54 | 0 | 0 |
| 26 | 38 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 29 | 69 |
| 64 | 0 | 0 | 0 |
| 36 | 9 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 41 | 65 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 41 | 65 |
| 64 | 0 | 0 | 0 |
| 36 | 9 | 0 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[27,38,0,0,0,46,0,0,0,0,46,35,0,0,0,27],[35,26,0,0,54,38,0,0,0,0,4,29,0,0,2,69],[64,36,0,0,0,9,0,0,0,0,8,41,0,0,0,65],[0,0,64,36,0,0,0,9,8,41,0,0,0,65,0,0] >;
C3×D4⋊D6 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes D_6
% in TeX
G:=Group("C3xD4:D6"); // GroupNames label
G:=SmallGroup(288,720);
// by ID
G=gap.SmallGroup(288,720);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,2524,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations