G = C6.1202+ 1+4 order 192 = 26·3
29th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1202+ 1+4, C4⋊C4⋊14D6, D6.5(C2×D4), C22⋊C4⋊14D6, (C22×S3)⋊8D4, (C22×C4)⋊25D6, C23⋊2D6⋊15C2, C12⋊D4⋊29C2, Dic3⋊D4⋊28C2, D6⋊D4⋊18C2, D6⋊C4⋊21C22, (C2×D4).162D6, C3⋊4(C23⋊3D4), C22.44(S3×D4), C6.82(C22×D4), D6.D4⋊26C2, C2.40(D4○D12), (C22×D12)⋊10C2, (C2×D12)⋊47C22, (C2×C6).197C24, (C2×C12).70C23, Dic3⋊C4⋊4C22, C22.D4⋊3S3, (S3×C23)⋊11C22, (C22×C12)⋊12C22, (C6×D4).135C22, C23.28D6⋊7C2, (C22×C6).32C23, C6.D4⋊29C22, C22.218(S3×C23), C23.210(C22×S3), (C22×S3).205C23, (C2×Dic3).101C23, (C2×S3×D4)⋊14C2, C2.55(C2×S3×D4), (S3×C2×C4)⋊20C22, (C2×C6).58(C2×D4), (C3×C4⋊C4)⋊24C22, (S3×C22⋊C4)⋊11C2, (C2×C3⋊D4)⋊18C22, (C3×C22⋊C4)⋊20C22, (C2×C4).189(C22×S3), (C3×C22.D4)⋊5C2, SmallGroup(192,1212)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.1202+ 1+4
G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=a3b-1, dbd-1=ebe=a3b, cd=dc, ce=ec, ede=a3b2d >
Subgroups: 1168 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22.D4, C22×D4, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×D12, S3×D4, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C23⋊3D4, S3×C22⋊C4, D6⋊D4, D6⋊D4, Dic3⋊D4, D6.D4, C12⋊D4, C23.28D6, C23⋊2D6, C3×C22.D4, C22×D12, C2×S3×D4, C6.1202+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, C23⋊3D4, C2×S3×D4, D4○D12, C6.1202+ 1+4
►On 48 points
(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 40 10 43)(2 41 11 44)(3 42 12 45)(4 37 7 46)(5 38 8 47)(6 39 9 48)(13 34 22 25)(14 35 23 26)(15 36 24 27)(16 31 19 28)(17 32 20 29)(18 33 21 30)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)(25 43)(26 48)(27 47)(28 46)(29 45)(30 44)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)
G:=sub<Sym(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), (1,40,10,43)(2,41,11,44)(3,42,12,45)(4,37,7,46)(5,38,8,47)(6,39,9,48)(13,34,22,25)(14,35,23,26)(15,36,24,27)(16,31,19,28)(17,32,20,29)(18,33,21,30), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23)(25,43)(26,48)(27,47)(28,46)(29,45)(30,44)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)>;
G:=Group( (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), (1,40,10,43)(2,41,11,44)(3,42,12,45)(4,37,7,46)(5,38,8,47)(6,39,9,48)(13,34,22,25)(14,35,23,26)(15,36,24,27)(16,31,19,28)(17,32,20,29)(18,33,21,30), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23)(25,43)(26,48)(27,47)(28,46)(29,45)(30,44)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38) );
G=PermutationGroup([[(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)], [(1,40,10,43),(2,41,11,44),(3,42,12,45),(4,37,7,46),(5,38,8,47),(6,39,9,48),(13,34,22,25),(14,35,23,26),(15,36,24,27),(16,31,19,28),(17,32,20,29),(18,33,21,30)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,22),(8,21),(9,20),(10,19),(11,24),(12,23),(25,43),(26,48),(27,47),(28,46),(29,45),(30,44),(31,37),(32,42),(33,41),(34,40),(35,39),(36,38)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | ··· | 4E | 4F | 4G | 4H | 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 | 2 | 2 | 3 | 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 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | 2+ 1+4 | S3×D4 | D4○D12 |
| kernel | C6.1202+ 1+4 | S3×C22⋊C4 | D6⋊D4 | Dic3⋊D4 | D6.D4 | C12⋊D4 | C23.28D6 | C23⋊2D6 | C3×C22.D4 | C22×D12 | C2×S3×D4 | C22.D4 | C22×S3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C6 | C22 | C2 |
| # reps | 1 | 1 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 2 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C6.1202+ 1+4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 9 | 9 |
| 0 | 0 | 7 | 3 | 0 | 9 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 8 | 12 | 0 |
| 0 | 0 | 10 | 8 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 7 | 6 | 3 | 6 |
| 0 | 0 | 6 | 0 | 3 | 10 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,9,0,10,6,0,0,9,9,7,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,8,10,0,0,0,1,8,8,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,6,3,7,6,0,0,10,7,6,0,0,0,0,0,3,3,0,0,0,0,6,10] >;
C6.1202+ 1+4 in GAP, Magma, Sage, TeX
C_6._{120}2_+^{1+4} % in TeX
G:=Group("C6.120ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1212);
// by ID
G=gap.SmallGroup(192,1212);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=e^2=1,d^2=a^3*b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c=a^3*b^-1,d*b*d^-1=e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=a^3*b^2*d>;
// generators/relations