direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8⋊3Dic3, C62.36D4, C32⋊12C4≀C2, (C3×D4)⋊2C12, (C3×Q8)⋊4C12, C12.9(C2×C12), (C3×D4)⋊5Dic3, (C4×Dic3)⋊2C6, Q8⋊4(C3×Dic3), D4⋊2(C3×Dic3), (C3×Q8)⋊7Dic3, C12.65(C3×D4), (D4×C32)⋊5C4, C4.Dic3⋊4C6, C4.3(C6×Dic3), (Q8×C32)⋊5C4, (Dic3×C12)⋊7C2, (C2×C12).319D6, (C3×C12).167D4, (C6×C12).51C22, C12.38(C2×Dic3), C12.148(C3⋊D4), C6.36(C6.D4), C3⋊3(C3×C4≀C2), (C2×C6).4(C3×D4), (C2×C4).40(S3×C6), (C3×C4○D4).9C6, C4○D4.5(C3×S3), C4.31(C3×C3⋊D4), (C2×C12).21(C2×C6), (C3×C12).45(C2×C4), (C3×C4○D4).20S3, C6.18(C3×C22⋊C4), C22.3(C3×C3⋊D4), (C2×C6).44(C3⋊D4), (C3×C4.Dic3)⋊20C2, C2.8(C3×C6.D4), (C32×C4○D4).1C2, (C3×C6).69(C22⋊C4), SmallGroup(288,271)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8⋊3Dic3
G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
Subgroups: 226 in 108 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×C6, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4≀C2, C3×Dic3, C3×C12, C3×C12, C62, C62, C4.Dic3, C4×Dic3, C4×C12, C3×M4(2), C3×C4○D4, C3×C4○D4, C3×C3⋊C8, C6×Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8⋊3Dic3, C3×C4≀C2, C3×C4.Dic3, Dic3×C12, C32×C4○D4, C3×Q8⋊3Dic3
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≀C2, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, Q8⋊3Dic3, C3×C4≀C2, C3×C6.D4, C3×Q8⋊3Dic3
►On 48 points
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 11 36 13)(2 12 31 14)(3 7 32 15)(4 8 33 16)(5 9 34 17)(6 10 35 18)(19 28 38 47)(20 29 39 48)(21 30 40 43)(22 25 41 44)(23 26 42 45)(24 27 37 46)
(1 16 36 8)(2 9 31 17)(3 18 32 10)(4 11 33 13)(5 14 34 12)(6 7 35 15)(19 41 38 22)(20 23 39 42)(21 37 40 24)(25 47 44 28)(26 29 45 48)(27 43 46 30)
(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 28 4 25)(2 27 5 30)(3 26 6 29)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 19 16 22)(14 24 17 21)(15 23 18 20)(31 46 34 43)(32 45 35 48)(33 44 36 47)
G:=sub<Sym(48)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,11,36,13)(2,12,31,14)(3,7,32,15)(4,8,33,16)(5,9,34,17)(6,10,35,18)(19,28,38,47)(20,29,39,48)(21,30,40,43)(22,25,41,44)(23,26,42,45)(24,27,37,46), (1,16,36,8)(2,9,31,17)(3,18,32,10)(4,11,33,13)(5,14,34,12)(6,7,35,15)(19,41,38,22)(20,23,39,42)(21,37,40,24)(25,47,44,28)(26,29,45,48)(27,43,46,30), (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), (1,28,4,25)(2,27,5,30)(3,26,6,29)(7,42,10,39)(8,41,11,38)(9,40,12,37)(13,19,16,22)(14,24,17,21)(15,23,18,20)(31,46,34,43)(32,45,35,48)(33,44,36,47)>;
G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,11,36,13)(2,12,31,14)(3,7,32,15)(4,8,33,16)(5,9,34,17)(6,10,35,18)(19,28,38,47)(20,29,39,48)(21,30,40,43)(22,25,41,44)(23,26,42,45)(24,27,37,46), (1,16,36,8)(2,9,31,17)(3,18,32,10)(4,11,33,13)(5,14,34,12)(6,7,35,15)(19,41,38,22)(20,23,39,42)(21,37,40,24)(25,47,44,28)(26,29,45,48)(27,43,46,30), (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), (1,28,4,25)(2,27,5,30)(3,26,6,29)(7,42,10,39)(8,41,11,38)(9,40,12,37)(13,19,16,22)(14,24,17,21)(15,23,18,20)(31,46,34,43)(32,45,35,48)(33,44,36,47) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,33,35),(32,34,36),(37,41,39),(38,42,40),(43,47,45),(44,48,46)], [(1,11,36,13),(2,12,31,14),(3,7,32,15),(4,8,33,16),(5,9,34,17),(6,10,35,18),(19,28,38,47),(20,29,39,48),(21,30,40,43),(22,25,41,44),(23,26,42,45),(24,27,37,46)], [(1,16,36,8),(2,9,31,17),(3,18,32,10),(4,11,33,13),(5,14,34,12),(6,7,35,15),(19,41,38,22),(20,23,39,42),(21,37,40,24),(25,47,44,28),(26,29,45,48),(27,43,46,30)], [(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)], [(1,28,4,25),(2,27,5,30),(3,26,6,29),(7,42,10,39),(8,41,11,38),(9,40,12,37),(13,19,16,22),(14,24,17,21),(15,23,18,20),(31,46,34,43),(32,45,35,48),(33,44,36,47)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | ··· | 6G | 6H | ··· | 6R | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | ··· | 12W | 12X | ··· | 12AE | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
72 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D4 | D4 | D6 | Dic3 | Dic3 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | C4≀C2 | S3×C6 | C3×Dic3 | C3×Dic3 | C3×C3⋊D4 | C3×C3⋊D4 | C3×C4≀C2 | Q8⋊3Dic3 | C3×Q8⋊3Dic3 |
| kernel | C3×Q8⋊3Dic3 | C3×C4.Dic3 | Dic3×C12 | C32×C4○D4 | Q8⋊3Dic3 | D4×C32 | Q8×C32 | C4.Dic3 | C4×Dic3 | C3×C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C12 | C2×C6 | C2×C6 | C32 | C2×C4 | D4 | Q8 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×Q8⋊3Dic3 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 9 |
| 14 | 59 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,0,27,0,0,0,0,1,0,0,0,0,72],[0,27,0,0,46,0,0,0,0,0,65,0,0,0,0,9],[14,14,0,0,59,14,0,0,0,0,0,72,0,0,1,0] >;
C3×Q8⋊3Dic3 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes_3{\rm Dic}_3 % in TeX
G:=Group("C3xQ8:3Dic3"); // GroupNames label
G:=SmallGroup(288,271);
// by ID
G=gap.SmallGroup(288,271);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,136,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations