direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic3⋊D4, C62.175C23, D6⋊1(C3×D4), D6⋊C4⋊10C6, (S3×C6)⋊11D4, (C2×D12)⋊3C6, C6.20(C6×D4), (C6×D12)⋊26C2, Dic3⋊C4⋊5C6, Dic3⋊2(C3×D4), C6.179(S3×D4), (C3×Dic3)⋊17D4, (C2×C12).266D6, C23.10(S3×C6), (C22×C6).28D6, C32⋊16(C4⋊D4), C6.118(C4○D12), (C6×C12).190C22, (C2×C62).51C22, (C6×Dic3).94C22, C2.9(C3×S3×D4), (S3×C2×C4)⋊11C6, (S3×C2×C12)⋊30C2, C3⋊1(C3×C4⋊D4), (C2×C3⋊D4)⋊2C6, (C2×C4).6(S3×C6), C6.8(C3×C4○D4), (C3×D6⋊C4)⋊29C2, (C6×C3⋊D4)⋊16C2, C22⋊C4⋊4(C3×S3), (C3×C22⋊C4)⋊6C6, C22.43(S3×C2×C6), (C3×C22⋊C4)⋊12S3, (C2×C12).53(C2×C6), C2.11(C3×C4○D12), (C3×C6).208(C2×D4), (S3×C2×C6).91C22, (C3×Dic3⋊C4)⋊24C2, (C3×C6).98(C4○D4), (C22×S3).4(C2×C6), (C22×C6).25(C2×C6), (C2×C6).30(C22×C6), (C32×C22⋊C4)⋊10C2, (C2×C6).308(C22×S3), (C2×Dic3).21(C2×C6), SmallGroup(288,655)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic3⋊D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b3c, ce=ec, ede=d-1 >
Subgroups: 546 in 205 conjugacy classes, 66 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, C2×C62, Dic3⋊D4, C3×C4⋊D4, C3×Dic3⋊C4, C3×D6⋊C4, C32×C22⋊C4, S3×C2×C12, C6×D12, C6×C3⋊D4, C3×Dic3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3×D4, C22×S3, C22×C6, C4⋊D4, S3×C6, C4○D12, S3×D4, C6×D4, C3×C4○D4, S3×C2×C6, Dic3⋊D4, C3×C4⋊D4, C3×C4○D12, C3×S3×D4, C3×Dic3⋊D4
►On 48 points
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(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 10 4 7)(2 9 5 12)(3 8 6 11)(13 43 16 46)(14 48 17 45)(15 47 18 44)(19 39 22 42)(20 38 23 41)(21 37 24 40)(25 35 28 32)(26 34 29 31)(27 33 30 36)
(1 33 14 40)(2 32 15 39)(3 31 16 38)(4 36 17 37)(5 35 18 42)(6 34 13 41)(7 30 45 21)(8 29 46 20)(9 28 47 19)(10 27 48 24)(11 26 43 23)(12 25 44 22)
(1 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 34)(14 33)(15 32)(16 31)(17 36)(18 35)(25 47)(26 46)(27 45)(28 44)(29 43)(30 48)
G:=sub<Sym(48)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (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), (1,10,4,7)(2,9,5,12)(3,8,6,11)(13,43,16,46)(14,48,17,45)(15,47,18,44)(19,39,22,42)(20,38,23,41)(21,37,24,40)(25,35,28,32)(26,34,29,31)(27,33,30,36), (1,33,14,40)(2,32,15,39)(3,31,16,38)(4,36,17,37)(5,35,18,42)(6,34,13,41)(7,30,45,21)(8,29,46,20)(9,28,47,19)(10,27,48,24)(11,26,43,23)(12,25,44,22), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,34)(14,33)(15,32)(16,31)(17,36)(18,35)(25,47)(26,46)(27,45)(28,44)(29,43)(30,48)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (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), (1,10,4,7)(2,9,5,12)(3,8,6,11)(13,43,16,46)(14,48,17,45)(15,47,18,44)(19,39,22,42)(20,38,23,41)(21,37,24,40)(25,35,28,32)(26,34,29,31)(27,33,30,36), (1,33,14,40)(2,32,15,39)(3,31,16,38)(4,36,17,37)(5,35,18,42)(6,34,13,41)(7,30,45,21)(8,29,46,20)(9,28,47,19)(10,27,48,24)(11,26,43,23)(12,25,44,22), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,34)(14,33)(15,32)(16,31)(17,36)(18,35)(25,47)(26,46)(27,45)(28,44)(29,43)(30,48) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46)], [(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)], [(1,10,4,7),(2,9,5,12),(3,8,6,11),(13,43,16,46),(14,48,17,45),(15,47,18,44),(19,39,22,42),(20,38,23,41),(21,37,24,40),(25,35,28,32),(26,34,29,31),(27,33,30,36)], [(1,33,14,40),(2,32,15,39),(3,31,16,38),(4,36,17,37),(5,35,18,42),(6,34,13,41),(7,30,45,21),(8,29,46,20),(9,28,47,19),(10,27,48,24),(11,26,43,23),(12,25,44,22)], [(1,40),(2,39),(3,38),(4,37),(5,42),(6,41),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,34),(14,33),(15,32),(16,31),(17,36),(18,35),(25,47),(26,46),(27,45),(28,44),(29,43),(30,48)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 6X | 6Y | 6Z | 6AA | 6AB | 6AC | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 12S | 12T | 12U | 12V | 12W | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3×S3 | C3×D4 | C3×D4 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 | S3×D4 | C3×S3×D4 |
| kernel | C3×Dic3⋊D4 | C3×Dic3⋊C4 | C3×D6⋊C4 | C32×C22⋊C4 | S3×C2×C12 | C6×D12 | C6×C3⋊D4 | Dic3⋊D4 | Dic3⋊C4 | D6⋊C4 | C3×C22⋊C4 | S3×C2×C4 | C2×D12 | C2×C3⋊D4 | C3×C22⋊C4 | C3×Dic3 | S3×C6 | C2×C12 | C22×C6 | C3×C6 | C22⋊C4 | Dic3 | D6 | C2×C4 | C23 | C6 | C6 | C2 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×Dic3⋊D4 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 5 | 0 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×Dic3⋊D4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_3\rtimes D_4 % in TeX
G:=Group("C3xDic3:D4"); // GroupNames label
G:=SmallGroup(288,655);
// by ID
G=gap.SmallGroup(288,655);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,176,1598,555,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations