G = C6.422+ 1+4 order 192 = 26·3
42nd non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.422+ 1+4, C4⋊C4⋊7D6, (C2×D4)⋊9D6, C4⋊D4⋊16S3, C22⋊C4⋊29D6, (C22×C4)⋊23D6, C23⋊2D6⋊12C2, D6⋊3D4⋊22C2, D6⋊Q8⋊15C2, D6.4(C4○D4), (C6×D4)⋊15C22, (C2×C6).157C24, (C2×C12).42C23, D6⋊C4.70C22, C4⋊Dic3⋊33C22, Dic3⋊4D4⋊10C2, C23.14D6⋊14C2, C23.12D6⋊18C2, C2.44(D4⋊6D6), Dic3⋊C4⋊29C22, (C22×C12)⋊41C22, C3⋊4(C22.32C24), (C4×Dic3)⋊24C22, (C2×Dic6)⋊26C22, C23.8D6⋊18C2, (C22×C6).24C23, C23.27(C22×S3), C23.11D6⋊20C2, C23.23D6⋊22C2, C6.D4⋊25C22, C23.28D6⋊22C2, (C22×S3).65C23, (S3×C23).49C22, C22.178(S3×C23), (C2×Dic3).76C23, (C22×Dic3)⋊21C22, (C4×C3⋊D4)⋊55C2, (S3×C22⋊C4)⋊7C2, C4⋊C4⋊S3⋊13C2, C2.41(S3×C4○D4), (C3×C4⋊D4)⋊19C2, (C3×C4⋊C4)⋊14C22, C6.154(C2×C4○D4), (S3×C2×C4).85C22, (C3×C22⋊C4)⋊16C22, (C2×C4).178(C22×S3), (C2×C3⋊D4).30C22, SmallGroup(192,1172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.422+ 1+4
G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a3b-1, bd=db, be=eb, dcd-1=ece=a3c, ede=a3b2d >
Subgroups: 720 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C22.32C24, C23.8D6, S3×C22⋊C4, Dic3⋊4D4, C23.11D6, D6⋊Q8, C4⋊C4⋊S3, C4×C3⋊D4, C23.28D6, C23.23D6, C23.12D6, C23⋊2D6, D6⋊3D4, C23.14D6, C3×C4⋊D4, C6.422+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.32C24, D4⋊6D6, S3×C4○D4, C6.422+ 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 25 10 34)(2 26 11 35)(3 27 12 36)(4 28 7 31)(5 29 8 32)(6 30 9 33)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 19 7 13)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)(25 46 31 40)(26 45 32 39)(27 44 33 38)(28 43 34 37)(29 48 35 42)(30 47 36 41)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
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), (1,25,10,34)(2,26,11,35)(3,27,12,36)(4,28,7,31)(5,29,8,32)(6,30,9,33)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,46,31,40)(26,45,32,39)(27,44,33,38)(28,43,34,37)(29,48,35,42)(30,47,36,41), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)>;
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), (1,25,10,34)(2,26,11,35)(3,27,12,36)(4,28,7,31)(5,29,8,32)(6,30,9,33)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,46,31,40)(26,45,32,39)(27,44,33,38)(28,43,34,37)(29,48,35,42)(30,47,36,41), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45) );
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)], [(1,25,10,34),(2,26,11,35),(3,27,12,36),(4,28,7,31),(5,29,8,32),(6,30,9,33),(13,40,22,43),(14,41,23,44),(15,42,24,45),(16,37,19,46),(17,38,20,47),(18,39,21,48)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,19,7,13),(2,24,8,18),(3,23,9,17),(4,22,10,16),(5,21,11,15),(6,20,12,14),(25,46,31,40),(26,45,32,39),(27,44,33,38),(28,43,34,37),(29,48,35,42),(30,47,36,41)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | D4⋊6D6 | S3×C4○D4 |
| kernel | C6.422+ 1+4 | C23.8D6 | S3×C22⋊C4 | Dic3⋊4D4 | C23.11D6 | D6⋊Q8 | C4⋊C4⋊S3 | C4×C3⋊D4 | C23.28D6 | C23.23D6 | C23.12D6 | C23⋊2D6 | D6⋊3D4 | C23.14D6 | C3×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 4 | 2 | 4 | 2 |
Matrix representation of C6.422+ 1+4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,11,4,0,0,0,0,9,2,0,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,1,12,0,0,0,0,0,12,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C6.422+ 1+4 in GAP, Magma, Sage, TeX
C_6._{42}2_+^{1+4} % in TeX
G:=Group("C6.42ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1172);
// by ID
G=gap.SmallGroup(192,1172);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,297,6278]);
// 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,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^3*b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
// generators/relations