metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊28D6, C6.762+ 1+4, (C4×S3)⋊5D4, (C2×D4)⋊12D6, C4⋊1D4⋊6S3, C4.34(S3×D4), D6.47(C2×D4), C12.65(C2×D4), C23⋊2D6⋊26C2, C12⋊3D4⋊26C2, (C4×C12)⋊26C22, D6⋊C4⋊34C22, (C6×D4)⋊32C22, C6.93(C22×D4), C42⋊7S3⋊25C2, C42⋊2S3⋊23C2, (C2×C6).259C24, Dic3.52(C2×D4), C23.14D6⋊36C2, C23.12D6⋊26C2, C2.80(D4⋊6D6), (C2×C12).635C23, Dic3⋊C4⋊71C22, C3⋊5(C22.29C24), (C2×Dic6)⋊34C22, (C4×Dic3)⋊39C22, (C22×C6).73C23, C23.75(C22×S3), (C2×D12).170C22, C6.D4⋊36C22, (S3×C23).72C22, C22.280(S3×C23), (C22×S3).227C23, (C2×Dic3).134C23, (C22×Dic3)⋊29C22, (C2×S3×D4)⋊19C2, C2.66(C2×S3×D4), (C3×C4⋊1D4)⋊6C2, (C2×D4⋊2S3)⋊20C2, (C2×C3⋊D4)⋊26C22, (S3×C2×C4).138C22, (C2×C4).213(C22×S3), SmallGroup(192,1274)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊28D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=b-1, dbd=a2b-1, dcd=c-1 >
Subgroups: 1040 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊1D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C4×C12, C2×Dic6, S3×C2×C4, C2×D12, S3×D4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C6×D4, S3×C23, C22.29C24, C42⋊2S3, C42⋊7S3, C23.12D6, C23⋊2D6, C23.14D6, C12⋊3D4, C3×C4⋊1D4, C2×S3×D4, C2×D4⋊2S3, C42⋊28D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, C22.29C24, C2×S3×D4, D4⋊6D6, C42⋊28D6
►On 48 points
(1 14 7 17)(2 18 8 15)(3 16 9 13)(4 20 11 23)(5 24 12 21)(6 22 10 19)(25 48 42 35)(26 36 37 43)(27 44 38 31)(28 32 39 45)(29 46 40 33)(30 34 41 47)
(1 39 5 42)(2 37 6 40)(3 41 4 38)(7 28 12 25)(8 26 10 29)(9 30 11 27)(13 34 23 44)(14 45 24 35)(15 36 19 46)(16 47 20 31)(17 32 21 48)(18 43 22 33)
(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 12)(2 11)(3 10)(4 8)(5 7)(6 9)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)(26 30)(27 29)(31 46)(32 45)(33 44)(34 43)(35 48)(36 47)(37 41)(38 40)
G:=sub<Sym(48)| (1,14,7,17)(2,18,8,15)(3,16,9,13)(4,20,11,23)(5,24,12,21)(6,22,10,19)(25,48,42,35)(26,36,37,43)(27,44,38,31)(28,32,39,45)(29,46,40,33)(30,34,41,47), (1,39,5,42)(2,37,6,40)(3,41,4,38)(7,28,12,25)(8,26,10,29)(9,30,11,27)(13,34,23,44)(14,45,24,35)(15,36,19,46)(16,47,20,31)(17,32,21,48)(18,43,22,33), (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), (1,12)(2,11)(3,10)(4,8)(5,7)(6,9)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20)(26,30)(27,29)(31,46)(32,45)(33,44)(34,43)(35,48)(36,47)(37,41)(38,40)>;
G:=Group( (1,14,7,17)(2,18,8,15)(3,16,9,13)(4,20,11,23)(5,24,12,21)(6,22,10,19)(25,48,42,35)(26,36,37,43)(27,44,38,31)(28,32,39,45)(29,46,40,33)(30,34,41,47), (1,39,5,42)(2,37,6,40)(3,41,4,38)(7,28,12,25)(8,26,10,29)(9,30,11,27)(13,34,23,44)(14,45,24,35)(15,36,19,46)(16,47,20,31)(17,32,21,48)(18,43,22,33), (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), (1,12)(2,11)(3,10)(4,8)(5,7)(6,9)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20)(26,30)(27,29)(31,46)(32,45)(33,44)(34,43)(35,48)(36,47)(37,41)(38,40) );
G=PermutationGroup([[(1,14,7,17),(2,18,8,15),(3,16,9,13),(4,20,11,23),(5,24,12,21),(6,22,10,19),(25,48,42,35),(26,36,37,43),(27,44,38,31),(28,32,39,45),(29,46,40,33),(30,34,41,47)], [(1,39,5,42),(2,37,6,40),(3,41,4,38),(7,28,12,25),(8,26,10,29),(9,30,11,27),(13,34,23,44),(14,45,24,35),(15,36,19,46),(16,47,20,31),(17,32,21,48),(18,43,22,33)], [(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)], [(1,12),(2,11),(3,10),(4,8),(5,7),(6,9),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20),(26,30),(27,29),(31,46),(32,45),(33,44),(34,43),(35,48),(36,47),(37,41),(38,40)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | 2+ 1+4 | S3×D4 | D4⋊6D6 |
| kernel | C42⋊28D6 | C42⋊2S3 | C42⋊7S3 | C23.12D6 | C23⋊2D6 | C23.14D6 | C12⋊3D4 | C3×C4⋊1D4 | C2×S3×D4 | C2×D4⋊2S3 | C4⋊1D4 | C4×S3 | C42 | C2×D4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C42⋊28D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 6 | 6 |
| 0 | 0 | 11 | 11 | 0 | 6 |
| 0 | 0 | 0 | 9 | 2 | 4 |
| 0 | 0 | 11 | 0 | 2 | 4 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 8 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 1 | 1 | 11 | 12 |
| 0 | 0 | 5 | 5 | 12 | 0 |
| 0 | 0 | 4 | 5 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 9 | 10 | 12 | 12 |
| 0 | 0 | 8 | 9 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 12 | 12 |
| 0 | 0 | 0 | 12 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,11,0,11,0,0,9,11,9,0,0,0,6,0,2,2,0,0,6,6,4,4],[1,8,0,0,0,0,3,12,0,0,0,0,0,0,0,1,5,4,0,0,0,1,5,5,0,0,12,11,12,12,0,0,1,12,0,0],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,0,12,9,8,0,0,1,1,10,9,0,0,0,0,12,1,0,0,0,0,12,0],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,12,12,0,12,0,0,0,0,12,0,0,0,0,0,12,1] >;
C42⋊28D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{28}D_6 % in TeX
G:=Group("C4^2:28D6"); // GroupNames label
G:=SmallGroup(192,1274);
// by ID
G=gap.SmallGroup(192,1274);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations