direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8○D12, C32⋊82- 1+4, C62.152C23, (S3×Q8)⋊8C6, C4○D12⋊9C6, D4⋊2S3⋊5C6, (C3×D4).50D6, D4.10(S3×C6), Q8.21(S3×C6), (C3×Q8).76D6, (C2×Dic6)⋊14C6, (C6×Dic6)⋊24C2, D12.14(C2×C6), (C2×C12).256D6, C6.81(S3×C23), (C3×C6).50C24, C6.13(C23×C6), D6.7(C22×C6), (S3×C6).35C23, C12.27(C22×C6), Dic6.14(C2×C6), C3⋊2(C3×2- 1+4), (S3×C12).37C22, (C6×C12).167C22, C12.178(C22×S3), (C3×C12).128C23, (C3×D12).48C22, Dic3.8(C22×C6), (C3×Dic3).36C23, (C3×Dic6).49C22, (D4×C32).30C22, (Q8×C32).32C22, (C6×Dic3).104C22, C4.34(S3×C2×C6), C3⋊D4.(C2×C6), (C3×S3×Q8)⋊12C2, C4○D4⋊8(C3×S3), (C3×C4○D4)⋊9C6, C22.4(S3×C2×C6), (C3×C4○D4)⋊13S3, (C4×S3).6(C2×C6), (C2×C4).23(S3×C6), C2.14(S3×C22×C6), (C3×C4○D12)⋊19C2, (C2×C12).49(C2×C6), (C3×D4).10(C2×C6), (C32×C4○D4)⋊7C2, (C2×C6).5(C22×C6), (C3×Q8).23(C2×C6), (C3×D4⋊2S3)⋊12C2, (C2×C6).24(C22×S3), (C3×C3⋊D4).4C22, (C2×Dic3).14(C2×C6), SmallGroup(288,1000)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8○D12
G = < a,b,c,d,e | a3=b4=e2=1, c2=d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >
Subgroups: 578 in 312 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×Q8, C4○D4, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, 2- 1+4, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, Q8×C32, Q8○D12, C3×2- 1+4, C6×Dic6, C3×C4○D12, C3×D4⋊2S3, C3×S3×Q8, C32×C4○D4, C3×Q8○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, 2- 1+4, S3×C6, S3×C23, C23×C6, S3×C2×C6, Q8○D12, C3×2- 1+4, S3×C22×C6, C3×Q8○D12
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 21 7 15)(2 22 8 16)(3 23 9 17)(4 24 10 18)(5 13 11 19)(6 14 12 20)(25 37 31 43)(26 38 32 44)(27 39 33 45)(28 40 34 46)(29 41 35 47)(30 42 36 48)
(1 18 7 24)(2 19 8 13)(3 20 9 14)(4 21 10 15)(5 22 11 16)(6 23 12 17)(25 40 31 46)(26 41 32 47)(27 42 33 48)(28 43 34 37)(29 44 35 38)(30 45 36 39)
(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,21,7,15)(2,22,8,16)(3,23,9,17)(4,24,10,18)(5,13,11,19)(6,14,12,20)(25,37,31,43)(26,38,32,44)(27,39,33,45)(28,40,34,46)(29,41,35,47)(30,42,36,48), (1,18,7,24)(2,19,8,13)(3,20,9,14)(4,21,10,15)(5,22,11,16)(6,23,12,17)(25,40,31,46)(26,41,32,47)(27,42,33,48)(28,43,34,37)(29,44,35,38)(30,45,36,39), (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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,48)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,21,7,15)(2,22,8,16)(3,23,9,17)(4,24,10,18)(5,13,11,19)(6,14,12,20)(25,37,31,43)(26,38,32,44)(27,39,33,45)(28,40,34,46)(29,41,35,47)(30,42,36,48), (1,18,7,24)(2,19,8,13)(3,20,9,14)(4,21,10,15)(5,22,11,16)(6,23,12,17)(25,40,31,46)(26,41,32,47)(27,42,33,48)(28,43,34,37)(29,44,35,38)(30,45,36,39), (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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,48) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,21,7,15),(2,22,8,16),(3,23,9,17),(4,24,10,18),(5,13,11,19),(6,14,12,20),(25,37,31,43),(26,38,32,44),(27,39,33,45),(28,40,34,46),(29,41,35,47),(30,42,36,48)], [(1,18,7,24),(2,19,8,13),(3,20,9,14),(4,21,10,15),(5,22,11,16),(6,23,12,17),(25,40,31,46),(26,41,32,47),(27,42,33,48),(28,43,34,37),(29,44,35,38),(30,45,36,39)], [(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)], [(1,27),(2,26),(3,25),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,48)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | ··· | 6K | 6L | ··· | 6T | 6U | 6V | 6W | 6X | 12A | ··· | 12N | 12O | ··· | 12W | 12X | ··· | 12AI |
| order | 1 | 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 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D6 | D6 | D6 | C3×S3 | S3×C6 | S3×C6 | S3×C6 | 2- 1+4 | Q8○D12 | C3×2- 1+4 | C3×Q8○D12 |
| kernel | C3×Q8○D12 | C6×Dic6 | C3×C4○D12 | C3×D4⋊2S3 | C3×S3×Q8 | C32×C4○D4 | Q8○D12 | C2×Dic6 | C4○D12 | D4⋊2S3 | S3×Q8 | C3×C4○D4 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C32 | C3 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 2 | 6 | 6 | 12 | 4 | 2 | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of C3×Q8○D12 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 0 | 8 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[0,8,0,0,8,0,0,0,0,0,0,8,0,0,8,0],[6,0,0,0,0,6,0,0,0,0,11,0,0,0,0,11],[0,0,7,0,0,0,0,7,2,0,0,0,0,2,0,0] >;
C3×Q8○D12 in GAP, Magma, Sage, TeX
C_3\times Q_8\circ D_{12} % in TeX
G:=Group("C3xQ8oD12"); // GroupNames label
G:=SmallGroup(288,1000);
// by ID
G=gap.SmallGroup(288,1000);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,344,555,268,1571,192,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=e^2=1,c^2=d^6=b^2,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,c*d=d*c,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations