metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊6D6, C4○D12⋊5C4, (C2×D12)⋊12C4, (C4×C12)⋊3C22, C4.83(C2×D12), C42⋊4S3⋊2C2, C42⋊C2⋊4S3, C4.10(D6⋊C4), (C2×Dic6)⋊12C4, D12.22(C2×C4), (C2×C12).144D4, C12.303(C2×D4), (C2×C4).147D12, (C22×C6).78D4, Dic6.23(C2×C4), (C22×C4).129D6, C12.23(C22⋊C4), C12.110(C22×C4), (C2×C12).794C23, C22.25(D6⋊C4), C3⋊2(C42⋊C22), C4○D12.38C22, C23.28(C3⋊D4), C4.Dic3⋊20C22, (C22×C12).154C22, C4.68(S3×C2×C4), (C2×C4).46(C4×S3), C2.20(C2×D6⋊C4), (C2×C12).94(C2×C4), (C2×C4○D12).8C2, (C2×C6).461(C2×D4), C6.47(C2×C22⋊C4), (C3×C42⋊C2)⋊4C2, (C2×C4).45(C3⋊D4), (C2×C4.Dic3)⋊10C2, C22.27(C2×C3⋊D4), (C2×C6).17(C22⋊C4), (C2×C4).708(C22×S3), SmallGroup(192,564)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊6D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 424 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C42⋊C22, C42⋊4S3, C2×C4.Dic3, C3×C42⋊C2, C2×C4○D12, C42⋊6D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C42⋊C22, C2×D6⋊C4, C42⋊6D6
►On 48 points
(1 37 16 28)(2 41 17 26)(3 39 18 30)(4 40 13 25)(5 38 14 29)(6 42 15 27)(7 43 22 34)(8 47 23 32)(9 45 24 36)(10 46 19 31)(11 44 20 35)(12 48 21 33)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 46 40 43)(38 47 41 44)(39 48 42 45)
(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)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 21)(22 24)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)
G:=sub<Sym(48)| (1,37,16,28)(2,41,17,26)(3,39,18,30)(4,40,13,25)(5,38,14,29)(6,42,15,27)(7,43,22,34)(8,47,23,32)(9,45,24,36)(10,46,19,31)(11,44,20,35)(12,48,21,33), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21)(25,34,28,31)(26,35,29,32)(27,36,30,33)(37,46,40,43)(38,47,41,44)(39,48,42,45), (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)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46)>;
G:=Group( (1,37,16,28)(2,41,17,26)(3,39,18,30)(4,40,13,25)(5,38,14,29)(6,42,15,27)(7,43,22,34)(8,47,23,32)(9,45,24,36)(10,46,19,31)(11,44,20,35)(12,48,21,33), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21)(25,34,28,31)(26,35,29,32)(27,36,30,33)(37,46,40,43)(38,47,41,44)(39,48,42,45), (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)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46) );
G=PermutationGroup([[(1,37,16,28),(2,41,17,26),(3,39,18,30),(4,40,13,25),(5,38,14,29),(6,42,15,27),(7,43,22,34),(8,47,23,32),(9,45,24,36),(10,46,19,31),(11,44,20,35),(12,48,21,33)], [(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,16,19),(14,23,17,20),(15,24,18,21),(25,34,28,31),(26,35,29,32),(27,36,30,33),(37,46,40,43),(38,47,41,44),(39,48,42,45)], [(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),(7,12),(8,11),(9,10),(13,18),(14,17),(15,16),(19,21),(22,24),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,45),(38,44),(39,43),(40,48),(41,47),(42,46)]])
42 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 | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | C42⋊C22 | C42⋊6D6 |
| kernel | C42⋊6D6 | C42⋊4S3 | C2×C4.Dic3 | C3×C42⋊C2 | C2×C4○D12 | C2×Dic6 | C2×D12 | C4○D12 | C42⋊C2 | C2×C12 | C22×C6 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C42⋊6D6 ►in GL4(𝔽73) generated by
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 43 | 13 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 14 | 7 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 66 | 7 |
G:=sub<GL(4,GF(73))| [0,0,43,60,0,0,13,30,72,0,0,0,0,72,0,0],[66,14,0,0,59,7,0,0,0,0,66,14,0,0,59,7],[72,1,0,0,72,0,0,0,0,0,1,72,0,0,1,0],[72,1,0,0,0,1,0,0,0,0,66,66,0,0,59,7] >;
C42⋊6D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_6D_6
% in TeX
G:=Group("C4^2:6D6"); // GroupNames label
G:=SmallGroup(192,564);
// by ID
G=gap.SmallGroup(192,564);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,1123,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations