metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊3D6, M4(2)⋊16D6, C4≀C2⋊5S3, (S3×D4)⋊4C4, (S3×Q8)⋊4C4, D4⋊2S3⋊4C4, Q8⋊3S3⋊4C4, C4○D4.35D6, D12.5(C2×C4), D4.11(C4×S3), (C4×S3).48D4, C4.201(S3×D4), Q8.16(C4×S3), C42⋊4S3⋊4C2, (C4×C12)⋊10C22, C42⋊2S3⋊9C2, C12.360(C2×D4), (S3×M4(2))⋊9C2, Dic6.5(C2×C4), D12⋊C4⋊10C2, C22.28(S3×D4), Q8⋊3Dic3⋊2C2, C12.18(C22×C4), D6.8(C22⋊C4), (C2×Dic3).37D4, (C4×Dic3)⋊3C22, (C22×S3).23D4, C4.Dic3⋊3C22, (C2×C12).261C23, C3⋊1(C42⋊C22), C4○D12.10C22, (C3×M4(2))⋊18C22, Dic3.14(C22⋊C4), C4.18(S3×C2×C4), (C3×C4≀C2)⋊10C2, (C4×S3).5(C2×C4), (S3×C4○D4).2C2, (C3×D4).5(C2×C4), (C2×C6).25(C2×D4), (C3×Q8).5(C2×C4), C2.26(S3×C22⋊C4), C6.25(C2×C22⋊C4), (S3×C2×C4).29C22, (C3×C4○D4).2C22, (C2×C4).368(C22×S3), SmallGroup(192,380)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊3D6
G = < a,b,c,d | a4=b4=c6=d2=1, cac-1=ab=ba, dad=ab-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 448 in 154 conjugacy classes, 51 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C4≀C2, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, S3×C8, C8⋊S3, C4.Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×M4(2), S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, C42⋊C22, C42⋊4S3, D12⋊C4, Q8⋊3Dic3, C3×C4≀C2, C42⋊2S3, S3×M4(2), S3×C4○D4, C42⋊3D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4, C42⋊C22, S3×C22⋊C4, C42⋊3D6
►On 48 points
(1 30 22 44)(2 28 20 48)(3 26 24 46)(4 13 38 27)(5 17 42 25)(6 15 40 29)(7 32 23 14)(8 36 21 18)(9 34 19 16)(10 43 37 33)(11 47 41 31)(12 45 39 35)
(1 38 9 41)(2 42 7 39)(3 40 8 37)(4 19 11 22)(5 23 12 20)(6 21 10 24)(13 16 47 44)(14 45 48 17)(15 18 43 46)(25 32 35 28)(26 29 36 33)(27 34 31 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 3)(4 6)(8 9)(10 11)(13 46)(14 45)(15 44)(16 43)(17 48)(18 47)(19 24)(20 23)(21 22)(25 32)(26 31)(27 36)(28 35)(29 34)(30 33)(37 38)(39 42)(40 41)
G:=sub<Sym(48)| (1,30,22,44)(2,28,20,48)(3,26,24,46)(4,13,38,27)(5,17,42,25)(6,15,40,29)(7,32,23,14)(8,36,21,18)(9,34,19,16)(10,43,37,33)(11,47,41,31)(12,45,39,35), (1,38,9,41)(2,42,7,39)(3,40,8,37)(4,19,11,22)(5,23,12,20)(6,21,10,24)(13,16,47,44)(14,45,48,17)(15,18,43,46)(25,32,35,28)(26,29,36,33)(27,34,31,30), (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), (1,3)(4,6)(8,9)(10,11)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,24)(20,23)(21,22)(25,32)(26,31)(27,36)(28,35)(29,34)(30,33)(37,38)(39,42)(40,41)>;
G:=Group( (1,30,22,44)(2,28,20,48)(3,26,24,46)(4,13,38,27)(5,17,42,25)(6,15,40,29)(7,32,23,14)(8,36,21,18)(9,34,19,16)(10,43,37,33)(11,47,41,31)(12,45,39,35), (1,38,9,41)(2,42,7,39)(3,40,8,37)(4,19,11,22)(5,23,12,20)(6,21,10,24)(13,16,47,44)(14,45,48,17)(15,18,43,46)(25,32,35,28)(26,29,36,33)(27,34,31,30), (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), (1,3)(4,6)(8,9)(10,11)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,24)(20,23)(21,22)(25,32)(26,31)(27,36)(28,35)(29,34)(30,33)(37,38)(39,42)(40,41) );
G=PermutationGroup([[(1,30,22,44),(2,28,20,48),(3,26,24,46),(4,13,38,27),(5,17,42,25),(6,15,40,29),(7,32,23,14),(8,36,21,18),(9,34,19,16),(10,43,37,33),(11,47,41,31),(12,45,39,35)], [(1,38,9,41),(2,42,7,39),(3,40,8,37),(4,19,11,22),(5,23,12,20),(6,21,10,24),(13,16,47,44),(14,45,48,17),(15,18,43,46),(25,32,35,28),(26,29,36,33),(27,34,31,30)], [(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)], [(1,3),(4,6),(8,9),(10,11),(13,46),(14,45),(15,44),(16,43),(17,48),(18,47),(19,24),(20,23),(21,22),(25,32),(26,31),(27,36),(28,35),(29,34),(30,33),(37,38),(39,42),(40,41)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12G | 12H | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 6 | 6 | 12 | 2 | 1 | 1 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 4 | 8 | 4 | 4 | 12 | 12 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4×S3 | C4×S3 | S3×D4 | S3×D4 | C42⋊C22 | C42⋊3D6 |
| kernel | C42⋊3D6 | C42⋊4S3 | D12⋊C4 | Q8⋊3Dic3 | C3×C4≀C2 | C42⋊2S3 | S3×M4(2) | S3×C4○D4 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 | C4≀C2 | C4×S3 | C2×Dic3 | C22×S3 | C42 | M4(2) | C4○D4 | D4 | Q8 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 |
Matrix representation of C42⋊3D6 ►in GL4(𝔽73) generated by
| 55 | 37 | 55 | 37 |
| 36 | 18 | 36 | 18 |
| 18 | 36 | 55 | 37 |
| 37 | 55 | 36 | 18 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 72 |
| 1 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 |
G:=sub<GL(4,GF(73))| [55,36,18,37,37,18,36,55,55,36,55,36,37,18,37,18],[0,0,1,0,0,0,0,1,72,0,0,0,0,72,0,0],[0,0,72,1,0,0,72,0,72,1,0,0,72,0,0,0],[0,0,1,0,0,0,1,72,1,0,0,0,1,72,0,0] >;
C42⋊3D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3D_6
% in TeX
G:=Group("C4^2:3D6"); // GroupNames label
G:=SmallGroup(192,380);
// by ID
G=gap.SmallGroup(192,380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,58,136,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a*b^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations