direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.10D4, (C2×C12).3C12, (C6×C12).12C4, (C3×C12).48D4, C12.10(C3×D4), (C6×Q8).27S3, (C6×Q8).10C6, (C2×C12).223D6, C62.99(C2×C4), (C2×C12).3Dic3, C4.Dic3.4C6, (C6×C12).50C22, C22.4(C6×Dic3), C12.102(C3⋊D4), C32⋊7(C4.10D4), C6.35(C6.D4), (Q8×C3×C6).2C2, (C2×C4).4(S3×C6), (C2×C4).(C3×Dic3), C4.15(C3×C3⋊D4), (C2×Q8).6(C3×S3), (C2×C12).20(C2×C6), (C2×C6).40(C2×C12), C3⋊2(C3×C4.10D4), C6.17(C3×C22⋊C4), (C2×C6).25(C2×Dic3), C2.7(C3×C6.D4), (C3×C4.Dic3).8C2, (C3×C6).68(C22⋊C4), SmallGroup(288,270)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.10D4
G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b9c3 >
Subgroups: 170 in 95 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, C12, C12, C2×C6, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C4.10D4, C3×C12, C3×C12, C62, C4.Dic3, C3×M4(2), C6×Q8, C6×Q8, C3×C3⋊C8, C6×C12, C6×C12, Q8×C32, C12.10D4, C3×C4.10D4, C3×C4.Dic3, Q8×C3×C6, C3×C12.10D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C4.10D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, C12.10D4, C3×C4.10D4, C3×C6.D4, C3×C12.10D4
►On 48 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(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 41 4 38 7 47 10 44)(2 40 5 37 8 46 11 43)(3 39 6 48 9 45 12 42)(13 36 16 33 19 30 22 27)(14 35 17 32 20 29 23 26)(15 34 18 31 21 28 24 25)
(1 36 10 33 7 30 4 27)(2 29 11 26 8 35 5 32)(3 34 12 31 9 28 6 25)(13 47 22 44 19 41 16 38)(14 40 23 37 20 46 17 43)(15 45 24 42 21 39 18 48)
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(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), (1,41,4,38,7,47,10,44)(2,40,5,37,8,46,11,43)(3,39,6,48,9,45,12,42)(13,36,16,33,19,30,22,27)(14,35,17,32,20,29,23,26)(15,34,18,31,21,28,24,25), (1,36,10,33,7,30,4,27)(2,29,11,26,8,35,5,32)(3,34,12,31,9,28,6,25)(13,47,22,44,19,41,16,38)(14,40,23,37,20,46,17,43)(15,45,24,42,21,39,18,48)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(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), (1,41,4,38,7,47,10,44)(2,40,5,37,8,46,11,43)(3,39,6,48,9,45,12,42)(13,36,16,33,19,30,22,27)(14,35,17,32,20,29,23,26)(15,34,18,31,21,28,24,25), (1,36,10,33,7,30,4,27)(2,29,11,26,8,35,5,32)(3,34,12,31,9,28,6,25)(13,47,22,44,19,41,16,38)(14,40,23,37,20,46,17,43)(15,45,24,42,21,39,18,48) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(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)], [(1,41,4,38,7,47,10,44),(2,40,5,37,8,46,11,43),(3,39,6,48,9,45,12,42),(13,36,16,33,19,30,22,27),(14,35,17,32,20,29,23,26),(15,34,18,31,21,28,24,25)], [(1,36,10,33,7,30,4,27),(2,29,11,26,8,35,5,32),(3,34,12,31,9,28,6,25),(13,47,22,44,19,41,16,38),(14,40,23,37,20,46,17,43),(15,45,24,42,21,39,18,48)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6M | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Dic3 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3×Dic3 | S3×C6 | C3×C3⋊D4 | C4.10D4 | C12.10D4 | C3×C4.10D4 | C3×C12.10D4 |
| kernel | C3×C12.10D4 | C3×C4.Dic3 | Q8×C3×C6 | C12.10D4 | C6×C12 | C4.Dic3 | C6×Q8 | C2×C12 | C6×Q8 | C3×C12 | C2×C12 | C2×C12 | C2×Q8 | C12 | C12 | C2×C4 | C2×C4 | C4 | C32 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C12.10D4 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 3 | 1 | 0 |
| 0 | 3 | 0 | 4 |
| 2 | 6 | 0 | 6 |
| 6 | 0 | 3 | 4 |
| 2 | 1 | 2 | 0 |
| 0 | 3 | 5 | 1 |
| 4 | 0 | 2 | 1 |
| 6 | 0 | 1 | 0 |
| 0 | 2 | 3 | 1 |
| 1 | 3 | 1 | 1 |
| 4 | 3 | 0 | 3 |
| 4 | 4 | 5 | 4 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,0,2,6,3,3,6,0,1,0,0,3,0,4,6,4],[2,0,4,6,1,3,0,0,2,5,2,1,0,1,1,0],[0,1,4,4,2,3,3,4,3,1,0,5,1,1,3,4] >;
C3×C12.10D4 in GAP, Magma, Sage, TeX
C_3\times C_{12}._{10}D_4 % in TeX
G:=Group("C3xC12.10D4"); // GroupNames label
G:=SmallGroup(288,270);
// by ID
G=gap.SmallGroup(288,270);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,344,850,136,2524,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^9*c^3>;
// generators/relations