metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊19D6, (S3×D4)⋊2C4, (C2×C8)⋊19D6, (C4×S3).3D4, D4.9(C4×S3), D12.1(C2×C4), C6.D8⋊5C2, C4.155(S3×D4), D4⋊C4⋊16S3, (C2×C24)⋊28C22, (C2×D4).131D6, D4⋊Dic3⋊4C2, C2.D24⋊23C2, C12.104(C2×D4), C2.2(D8⋊S3), C2.1(Q8⋊3D6), C12.5(C22×C4), C22.69(S3×D4), C6.29(C8⋊C22), C4⋊Dic3⋊18C22, (C6×D4).31C22, (C22×S3).68D4, (C2×C12).210C23, D6.11(C22⋊C4), (C2×Dic3).139D4, (C2×D12).48C22, C3⋊1(C23.37D4), Dic3.7(C22⋊C4), C4.5(S3×C2×C4), (C2×S3×D4).4C2, (C2×C3⋊C8)⋊1C22, C4⋊C4⋊7S3⋊1C2, (C3×C4⋊C4)⋊2C22, (C4×S3).1(C2×C4), (C3×D4).3(C2×C4), (C2×C8⋊S3)⋊15C2, (S3×C2×C4).5C22, (C2×C6).223(C2×D4), C2.18(S3×C22⋊C4), C6.17(C2×C22⋊C4), (C3×D4⋊C4)⋊24C2, (C2×C4).317(C22×S3), SmallGroup(192,329)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊19D6
G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab-1, dcd=c-1 >
Subgroups: 680 in 190 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, D4⋊C4, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C8⋊S3, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, S3×D4, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C23.37D4, C6.D8, C2.D24, D4⋊Dic3, C3×D4⋊C4, C4⋊C4⋊7S3, C2×C8⋊S3, C2×S3×D4, C4⋊C4⋊19D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C8⋊C22, S3×C2×C4, S3×D4, C23.37D4, S3×C22⋊C4, D8⋊S3, Q8⋊3D6, C4⋊C4⋊19D6
►On 48 points
(1 32 19 8)(2 9 20 33)(3 34 21 10)(4 11 22 35)(5 36 23 12)(6 7 24 31)(13 16 40 37)(14 38 41 17)(15 18 42 39)(25 28 46 43)(26 44 47 29)(27 30 48 45)
(1 25 4 42)(2 40 5 29)(3 27 6 38)(7 14 34 45)(8 28 35 39)(9 16 36 47)(10 30 31 41)(11 18 32 43)(12 26 33 37)(13 23 44 20)(15 19 46 22)(17 21 48 24)
(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 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 37)(14 42)(15 41)(16 40)(17 39)(18 38)(19 24)(20 23)(21 22)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)(31 32)(33 36)(34 35)
G:=sub<Sym(48)| (1,32,19,8)(2,9,20,33)(3,34,21,10)(4,11,22,35)(5,36,23,12)(6,7,24,31)(13,16,40,37)(14,38,41,17)(15,18,42,39)(25,28,46,43)(26,44,47,29)(27,30,48,45), (1,25,4,42)(2,40,5,29)(3,27,6,38)(7,14,34,45)(8,28,35,39)(9,16,36,47)(10,30,31,41)(11,18,32,43)(12,26,33,37)(13,23,44,20)(15,19,46,22)(17,21,48,24), (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), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,37)(14,42)(15,41)(16,40)(17,39)(18,38)(19,24)(20,23)(21,22)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)(31,32)(33,36)(34,35)>;
G:=Group( (1,32,19,8)(2,9,20,33)(3,34,21,10)(4,11,22,35)(5,36,23,12)(6,7,24,31)(13,16,40,37)(14,38,41,17)(15,18,42,39)(25,28,46,43)(26,44,47,29)(27,30,48,45), (1,25,4,42)(2,40,5,29)(3,27,6,38)(7,14,34,45)(8,28,35,39)(9,16,36,47)(10,30,31,41)(11,18,32,43)(12,26,33,37)(13,23,44,20)(15,19,46,22)(17,21,48,24), (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), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,37)(14,42)(15,41)(16,40)(17,39)(18,38)(19,24)(20,23)(21,22)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)(31,32)(33,36)(34,35) );
G=PermutationGroup([[(1,32,19,8),(2,9,20,33),(3,34,21,10),(4,11,22,35),(5,36,23,12),(6,7,24,31),(13,16,40,37),(14,38,41,17),(15,18,42,39),(25,28,46,43),(26,44,47,29),(27,30,48,45)], [(1,25,4,42),(2,40,5,29),(3,27,6,38),(7,14,34,45),(8,28,35,39),(9,16,36,47),(10,30,31,41),(11,18,32,43),(12,26,33,37),(13,23,44,20),(15,19,46,22),(17,21,48,24)], [(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)], [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,37),(14,42),(15,41),(16,40),(17,39),(18,38),(19,24),(20,23),(21,22),(25,45),(26,44),(27,43),(28,48),(29,47),(30,46),(31,32),(33,36),(34,35)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4×S3 | C8⋊C22 | S3×D4 | S3×D4 | D8⋊S3 | Q8⋊3D6 |
| kernel | C4⋊C4⋊19D6 | C6.D8 | C2.D24 | D4⋊Dic3 | C3×D4⋊C4 | C4⋊C4⋊7S3 | C2×C8⋊S3 | C2×S3×D4 | S3×D4 | D4⋊C4 | C4×S3 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C4⋊C4⋊19D6 ►in GL6(𝔽73)
| 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C4⋊C4⋊19D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{19}D_6 % in TeX
G:=Group("C4:C4:19D6"); // GroupNames label
G:=SmallGroup(192,329);
// by ID
G=gap.SmallGroup(192,329);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,58,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations