G = C6.612+ 1+4 order 192 = 26·3
61st non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.612+ 1+4, C4⋊C4⋊29D6, C22⋊C4⋊16D6, (C2×D4).98D6, (C22×C4)⋊27D6, D6⋊D4⋊21C2, C23⋊2D6⋊17C2, D6⋊3D4⋊30C2, Dic3⋊D4⋊32C2, C12⋊7D4⋊21C2, D6⋊C4⋊28C22, C4.D12⋊31C2, D6.6(C4○D4), Dic3⋊5D4⋊33C2, D6.D4⋊29C2, C2.42(D4○D12), (C2×C6).201C24, C4⋊Dic3⋊15C22, C22.D4⋊6S3, C2.63(D4⋊6D6), (C2×C12).179C23, Dic3⋊C4⋊54C22, (C22×C12)⋊18C22, C3⋊7(C22.32C24), (C2×Dic6)⋊29C22, (C4×Dic3)⋊33C22, (C6×D4).139C22, (C2×D12).32C22, C23.8D6⋊30C2, C23.11D6⋊32C2, C6.D4⋊53C22, (S3×C23).58C22, (C22×S3).85C23, C22.222(S3×C23), (C22×C6).221C23, C23.139(C22×S3), (C2×Dic3).244C23, (C4×C3⋊D4)⋊6C2, (S3×C2×C4)⋊23C22, C4⋊C4⋊S3⋊27C2, C2.63(S3×C4○D4), (C3×C4⋊C4)⋊27C22, (S3×C22⋊C4)⋊13C2, C6.175(C2×C4○D4), (C2×C3⋊D4)⋊20C22, (C2×C4).64(C22×S3), (C3×C22⋊C4)⋊23C22, (C3×C22.D4)⋊9C2, SmallGroup(192,1216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.612+ 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, dbd-1=ebe=a3b, dcd-1=ece=a3c, ede=a3b2d >
Subgroups: 752 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, D12, 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, C22⋊Q8, C22.D4, 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, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C22.32C24, C23.8D6, S3×C22⋊C4, D6⋊D4, Dic3⋊D4, C23.11D6, Dic3⋊5D4, D6.D4, C4.D12, C4⋊C4⋊S3, C4×C3⋊D4, C12⋊7D4, C23⋊2D6, D6⋊3D4, C3×C22.D4, C6.612+ 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, D4○D12, C6.612+ 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 43 31 37)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)
(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 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
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)(25,43,31,37)(26,48,32,42)(27,47,33,41)(28,46,34,40)(29,45,35,39)(30,44,36,38), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)>;
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)(25,43,31,37)(26,48,32,42)(27,47,33,41)(28,46,34,40)(29,45,35,39)(30,44,36,38), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48) );
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),(25,43,31,37),(26,48,32,42),(27,47,33,41),(28,46,34,40),(29,45,35,39),(30,44,36,38)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | D4○D12 |
| kernel | C6.612+ 1+4 | C23.8D6 | S3×C22⋊C4 | D6⋊D4 | Dic3⋊D4 | C23.11D6 | Dic3⋊5D4 | D6.D4 | C4.D12 | C4⋊C4⋊S3 | C4×C3⋊D4 | C12⋊7D4 | C23⋊2D6 | D6⋊3D4 | C3×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of C6.612+ 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 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 6 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 6 | 10 | 3 |
| 0 | 0 | 7 | 10 | 10 | 0 |
| 0 | 0 | 11 | 0 | 9 | 7 |
| 0 | 0 | 0 | 11 | 6 | 3 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 4 | 6 |
| 0 | 0 | 0 | 1 | 7 | 10 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 | 5 | 8 |
| 0 | 0 | 9 | 2 | 0 | 8 |
| 0 | 0 | 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 6 | 3 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[5,6,0,0,0,0,0,8,0,0,0,0,0,0,4,7,11,0,0,0,6,10,0,11,0,0,10,10,9,6,0,0,3,0,7,3],[12,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,4,7,12,0,0,0,6,10,0,12],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,11,9,0,0,0,0,11,2,0,0,0,0,5,0,11,9,0,0,8,8,11,2],[6,10,0,0,0,0,3,7,0,0,0,0,0,0,2,4,0,0,0,0,9,11,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;
C6.612+ 1+4 in GAP, Magma, Sage, TeX
C_6._{61}2_+^{1+4} % in TeX
G:=Group("C6.61ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1216);
// by ID
G=gap.SmallGroup(192,1216);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,6278]);
// 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,d*b*d^-1=e*b*e=a^3*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
// generators/relations