metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊28D6, D6⋊5(C4○D4), C22⋊C4⋊31D6, D6⋊D4⋊20C2, Dic3⋊D4⋊31C2, C23⋊2D6⋊16C2, D6⋊C4⋊27C22, D6⋊Q8⋊29C2, (C2×Dic3)⋊21D4, (C2×D4).165D6, Dic3⋊5D4⋊32C2, C22.45(S3×D4), C6.85(C22×D4), (C2×C6).200C24, (C2×C12).73C23, Dic3.49(C2×D4), (C22×C4).338D6, Dic3⋊4D4⋊20C2, Dic3⋊C4⋊23C22, C3⋊6(C22.19C24), (C2×Dic6)⋊28C22, (C4×Dic3)⋊32C22, (C6×D4).138C22, C22.D4⋊18S3, (C22×C6).35C23, C23.37(C22×S3), (C2×D12).156C22, C23.16D6⋊13C2, C23.28D6⋊21C2, C22.221(S3×C23), (S3×C23).110C22, (C22×S3).208C23, (C22×C12).368C22, (C2×Dic3).104C23, C6.D4.43C22, (C22×Dic3)⋊25C22, C2.58(C2×S3×D4), (S3×C2×C4)⋊22C22, (S3×C22×C4)⋊24C2, C2.62(S3×C4○D4), (C2×C6).61(C2×D4), (C3×C4⋊C4)⋊26C22, C6.174(C2×C4○D4), (C2×D4⋊2S3)⋊17C2, (C2×C3⋊D4)⋊19C22, (C2×C4).63(C22×S3), (C3×C22⋊C4)⋊22C22, (C3×C22.D4)⋊8C2, SmallGroup(192,1215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊28D6
G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=dad=a-1, cac-1=ab2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
Subgroups: 880 in 330 conjugacy classes, 107 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, 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, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C23×C4, C2×C4○D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×C2×C4, C2×D12, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C22.19C24, C23.16D6, Dic3⋊4D4, D6⋊D4, Dic3⋊D4, Dic3⋊5D4, D6⋊Q8, C23.28D6, C23⋊2D6, C3×C22.D4, S3×C22×C4, C2×D4⋊2S3, C4⋊C4⋊28D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, S3×D4, S3×C23, C22.19C24, C2×S3×D4, S3×C4○D4, C4⋊C4⋊28D6
►On 48 points
(1 41 10 38)(2 47 11 44)(3 37 12 40)(4 43 7 46)(5 39 8 42)(6 45 9 48)(13 27 16 36)(14 34 17 25)(15 29 18 32)(19 28 22 31)(20 35 23 26)(21 30 24 33)
(1 17 7 22)(2 23 8 18)(3 13 9 24)(4 19 10 14)(5 15 11 20)(6 21 12 16)(25 43 31 38)(26 39 32 44)(27 45 33 40)(28 41 34 46)(29 47 35 42)(30 37 36 48)
(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 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 43)(38 48)(39 47)(40 46)(41 45)(42 44)
G:=sub<Sym(48)| (1,41,10,38)(2,47,11,44)(3,37,12,40)(4,43,7,46)(5,39,8,42)(6,45,9,48)(13,27,16,36)(14,34,17,25)(15,29,18,32)(19,28,22,31)(20,35,23,26)(21,30,24,33), (1,17,7,22)(2,23,8,18)(3,13,9,24)(4,19,10,14)(5,15,11,20)(6,21,12,16)(25,43,31,38)(26,39,32,44)(27,45,33,40)(28,41,34,46)(29,47,35,42)(30,37,36,48), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,43)(38,48)(39,47)(40,46)(41,45)(42,44)>;
G:=Group( (1,41,10,38)(2,47,11,44)(3,37,12,40)(4,43,7,46)(5,39,8,42)(6,45,9,48)(13,27,16,36)(14,34,17,25)(15,29,18,32)(19,28,22,31)(20,35,23,26)(21,30,24,33), (1,17,7,22)(2,23,8,18)(3,13,9,24)(4,19,10,14)(5,15,11,20)(6,21,12,16)(25,43,31,38)(26,39,32,44)(27,45,33,40)(28,41,34,46)(29,47,35,42)(30,37,36,48), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,43)(38,48)(39,47)(40,46)(41,45)(42,44) );
G=PermutationGroup([[(1,41,10,38),(2,47,11,44),(3,37,12,40),(4,43,7,46),(5,39,8,42),(6,45,9,48),(13,27,16,36),(14,34,17,25),(15,29,18,32),(19,28,22,31),(20,35,23,26),(21,30,24,33)], [(1,17,7,22),(2,23,8,18),(3,13,9,24),(4,19,10,14),(5,15,11,20),(6,21,12,16),(25,43,31,38),(26,39,32,44),(27,45,33,40),(28,41,34,46),(29,47,35,42),(30,37,36,48)], [(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)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34),(37,43),(38,48),(39,47),(40,46),(41,45),(42,44)]])
42 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 | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | S3×D4 | S3×C4○D4 |
| kernel | C4⋊C4⋊28D6 | C23.16D6 | Dic3⋊4D4 | D6⋊D4 | Dic3⋊D4 | Dic3⋊5D4 | D6⋊Q8 | C23.28D6 | C23⋊2D6 | C3×C22.D4 | S3×C22×C4 | C2×D4⋊2S3 | C22.D4 | C2×Dic3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 2 | 1 | 1 | 8 | 2 | 4 |
Matrix representation of C4⋊C4⋊28D6 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 11 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,5,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,11,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
C4⋊C4⋊28D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{28}D_6 % in TeX
G:=Group("C4:C4:28D6"); // GroupNames label
G:=SmallGroup(192,1215);
// by ID
G=gap.SmallGroup(192,1215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,1123,346,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations