direct product, metabelian, supersoluble, monomial
Aliases: C6×D4.S3, C62.123D4, (C6×D4).3C6, D4.7(S3×C6), (C3×C6)⋊8SD16, C3⋊3(C6×SD16), C6⋊2(C3×SD16), C6.46(C6×D4), (C2×Dic6)⋊9C6, Dic6⋊6(C2×C6), (C6×D4).26S3, (C3×D4).47D6, (C3×C12).86D4, C12.16(C3×D4), (C6×Dic6)⋊14C2, (C2×C12).324D6, C32⋊17(C2×SD16), C12.88(C3⋊D4), C12.13(C22×C6), (C3×C12).84C23, (C6×C12).119C22, C12.164(C22×S3), (C3×Dic6)⋊23C22, (D4×C32).23C22, (C2×C3⋊C8)⋊5C6, C3⋊C8⋊8(C2×C6), (C6×C3⋊C8)⋊21C2, C4.13(S3×C2×C6), (D4×C3×C6).5C2, C4.6(C3×C3⋊D4), (C3×C3⋊C8)⋊39C22, (C2×C4).48(S3×C6), (C3×D4).7(C2×C6), (C2×D4).4(C3×S3), (C2×C6).49(C3×D4), C2.10(C6×C3⋊D4), (C2×C12).30(C2×C6), (C3×C6).256(C2×D4), C6.147(C2×C3⋊D4), C22.22(C3×C3⋊D4), (C2×C6).115(C3⋊D4), SmallGroup(288,704)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D4.S3
G = < a,b,c,d,e | a6=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 362 in 163 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C2×SD16, C3×Dic3, C3×C12, C62, C62, C2×C3⋊C8, D4.S3, C2×C24, C3×SD16, C2×Dic6, C6×D4, C6×D4, C6×Q8, C3×C3⋊C8, C3×Dic6, C3×Dic6, C6×Dic3, C6×C12, D4×C32, D4×C32, C2×C62, C2×D4.S3, C6×SD16, C6×C3⋊C8, C3×D4.S3, C6×Dic6, D4×C3×C6, C6×D4.S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, SD16, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C2×SD16, S3×C6, D4.S3, C3×SD16, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, C2×D4.S3, C6×SD16, C3×D4.S3, C6×C3⋊D4, C6×D4.S3
►On 48 points
(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 29 19 7)(2 30 20 8)(3 25 21 9)(4 26 22 10)(5 27 23 11)(6 28 24 12)(13 35 40 47)(14 36 41 48)(15 31 42 43)(16 32 37 44)(17 33 38 45)(18 34 39 46)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 40)(38 41)(39 42)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(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 46 19 34)(2 47 20 35)(3 48 21 36)(4 43 22 31)(5 44 23 32)(6 45 24 33)(7 18 29 39)(8 13 30 40)(9 14 25 41)(10 15 26 42)(11 16 27 37)(12 17 28 38)
G:=sub<Sym(48)| (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), (1,29,19,7)(2,30,20,8)(3,25,21,9)(4,26,22,10)(5,27,23,11)(6,28,24,12)(13,35,40,47)(14,36,41,48)(15,31,42,43)(16,32,37,44)(17,33,38,45)(18,34,39,46), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,16)(14,17)(15,18)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(37,40)(38,41)(39,42), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(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), (1,46,19,34)(2,47,20,35)(3,48,21,36)(4,43,22,31)(5,44,23,32)(6,45,24,33)(7,18,29,39)(8,13,30,40)(9,14,25,41)(10,15,26,42)(11,16,27,37)(12,17,28,38)>;
G:=Group( (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), (1,29,19,7)(2,30,20,8)(3,25,21,9)(4,26,22,10)(5,27,23,11)(6,28,24,12)(13,35,40,47)(14,36,41,48)(15,31,42,43)(16,32,37,44)(17,33,38,45)(18,34,39,46), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,16)(14,17)(15,18)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(37,40)(38,41)(39,42), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(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), (1,46,19,34)(2,47,20,35)(3,48,21,36)(4,43,22,31)(5,44,23,32)(6,45,24,33)(7,18,29,39)(8,13,30,40)(9,14,25,41)(10,15,26,42)(11,16,27,37)(12,17,28,38) );
G=PermutationGroup([[(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)], [(1,29,19,7),(2,30,20,8),(3,25,21,9),(4,26,22,10),(5,27,23,11),(6,28,24,12),(13,35,40,47),(14,36,41,48),(15,31,42,43),(16,32,37,44),(17,33,38,45),(18,34,39,46)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,16),(14,17),(15,18),(19,26),(20,27),(21,28),(22,29),(23,30),(24,25),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(37,40),(38,41),(39,42)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(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)], [(1,46,19,34),(2,47,20,35),(3,48,21,36),(4,43,22,31),(5,44,23,32),(6,45,24,33),(7,18,29,39),(8,13,30,40),(9,14,25,41),(10,15,26,42),(11,16,27,37),(12,17,28,38)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6AE | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
72 irreducible representations
| dim | 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 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | SD16 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | C3×SD16 | C3×C3⋊D4 | C3×C3⋊D4 | D4.S3 | C3×D4.S3 |
| kernel | C6×D4.S3 | C6×C3⋊C8 | C3×D4.S3 | C6×Dic6 | D4×C3×C6 | C2×D4.S3 | C2×C3⋊C8 | D4.S3 | C2×Dic6 | C6×D4 | C6×D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×C6 | C2×D4 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | D4 | C6 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 2 | 4 |
Matrix representation of C6×D4.S3 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 55 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 62 | 15 | 0 | 0 | 0 | 0 |
| 65 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 6 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 51 |
| 0 | 0 | 0 | 0 | 63 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,55,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[62,65,0,0,0,0,15,11,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,0,0,0,63,0,0,0,0,51,0] >;
C6×D4.S3 in GAP, Magma, Sage, TeX
C_6\times D_4.S_3
% in TeX
G:=Group("C6xD4.S3"); // GroupNames label
G:=SmallGroup(288,704);
// by ID
G=gap.SmallGroup(288,704);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,2524,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations