G = C6.1212+ 1+4 order 192 = 26·3
30th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1212+ 1+4, C4⋊C4⋊15D6, C3⋊D4⋊6D4, C3⋊8(D4⋊5D4), C22⋊C4⋊15D6, D6.22(C2×D4), (C22×C4)⋊26D6, D6⋊10(C4○D4), C12⋊D4⋊30C2, D6⋊D4⋊19C2, Dic3⋊D4⋊29C2, D6⋊C4⋊22C22, D6⋊Q8⋊27C2, (C2×D4).163D6, Dic3⋊5D4⋊31C2, C6.83(C22×D4), C22.12(S3×D4), D6.D4⋊27C2, C2.41(D4○D12), (C2×D12)⋊26C22, (C2×C6).198C24, (C2×C12).71C23, Dic3.26(C2×D4), C22.D4⋊4S3, C23.14D6⋊20C2, Dic3⋊4D4⋊18C2, Dic3⋊C4⋊22C22, (C22×C12)⋊17C22, (C2×Dic6)⋊55C22, (C4×Dic3)⋊31C22, (C6×D4).136C22, (C22×C6).33C23, C23.36(C22×S3), C23.11D6⋊31C2, C6.D4⋊30C22, (S3×C23).57C22, C22.219(S3×C23), (C22×S3).206C23, (C2×Dic3).102C23, (C22×Dic3)⋊24C22, (C2×S3×D4)⋊15C2, C2.56(C2×S3×D4), (C2×D6⋊C4)⋊23C2, (S3×C2×C4)⋊21C22, C2.60(S3×C4○D4), (C2×C6).59(C2×D4), (C2×C4○D12)⋊11C2, (C3×C4⋊C4)⋊25C22, (S3×C22⋊C4)⋊12C2, C6.172(C2×C4○D4), (C2×C3⋊D4)⋊41C22, (C2×C4).61(C22×S3), (C3×C22⋊C4)⋊21C22, (C3×C22.D4)⋊6C2, SmallGroup(192,1213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.1212+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
Subgroups: 1024 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, 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, C3⋊D4, 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, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, D4⋊5D4, S3×C22⋊C4, Dic3⋊4D4, D6⋊D4, Dic3⋊D4, C23.11D6, Dic3⋊5D4, D6.D4, C12⋊D4, D6⋊Q8, C2×D6⋊C4, C23.14D6, C3×C22.D4, C2×C4○D12, C2×S3×D4, C6.1212+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, S3×D4, S3×C23, D4⋊5D4, C2×S3×D4, S3×C4○D4, D4○D12, C6.1212+ 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 29 13 22)(2 30 14 23)(3 25 15 24)(4 26 16 19)(5 27 17 20)(6 28 18 21)(7 35 44 42)(8 36 45 37)(9 31 46 38)(10 32 47 39)(11 33 48 40)(12 34 43 41)
(1 29 4 26)(2 28 5 25)(3 27 6 30)(7 37 10 40)(8 42 11 39)(9 41 12 38)(13 22 16 19)(14 21 17 24)(15 20 18 23)(31 46 34 43)(32 45 35 48)(33 44 36 47)
(1 31 13 38)(2 36 14 37)(3 35 15 42)(4 34 16 41)(5 33 17 40)(6 32 18 39)(7 25 44 24)(8 30 45 23)(9 29 46 22)(10 28 47 21)(11 27 48 20)(12 26 43 19)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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), (1,29,13,22)(2,30,14,23)(3,25,15,24)(4,26,16,19)(5,27,17,20)(6,28,18,21)(7,35,44,42)(8,36,45,37)(9,31,46,38)(10,32,47,39)(11,33,48,40)(12,34,43,41), (1,29,4,26)(2,28,5,25)(3,27,6,30)(7,37,10,40)(8,42,11,39)(9,41,12,38)(13,22,16,19)(14,21,17,24)(15,20,18,23)(31,46,34,43)(32,45,35,48)(33,44,36,47), (1,31,13,38)(2,36,14,37)(3,35,15,42)(4,34,16,41)(5,33,17,40)(6,32,18,39)(7,25,44,24)(8,30,45,23)(9,29,46,22)(10,28,47,21)(11,27,48,20)(12,26,43,19), (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)>;
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), (1,29,13,22)(2,30,14,23)(3,25,15,24)(4,26,16,19)(5,27,17,20)(6,28,18,21)(7,35,44,42)(8,36,45,37)(9,31,46,38)(10,32,47,39)(11,33,48,40)(12,34,43,41), (1,29,4,26)(2,28,5,25)(3,27,6,30)(7,37,10,40)(8,42,11,39)(9,41,12,38)(13,22,16,19)(14,21,17,24)(15,20,18,23)(31,46,34,43)(32,45,35,48)(33,44,36,47), (1,31,13,38)(2,36,14,37)(3,35,15,42)(4,34,16,41)(5,33,17,40)(6,32,18,39)(7,25,44,24)(8,30,45,23)(9,29,46,22)(10,28,47,21)(11,27,48,20)(12,26,43,19), (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48) );
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)], [(1,29,13,22),(2,30,14,23),(3,25,15,24),(4,26,16,19),(5,27,17,20),(6,28,18,21),(7,35,44,42),(8,36,45,37),(9,31,46,38),(10,32,47,39),(11,33,48,40),(12,34,43,41)], [(1,29,4,26),(2,28,5,25),(3,27,6,30),(7,37,10,40),(8,42,11,39),(9,41,12,38),(13,22,16,19),(14,21,17,24),(15,20,18,23),(31,46,34,43),(32,45,35,48),(33,44,36,47)], [(1,31,13,38),(2,36,14,37),(3,35,15,42),(4,34,16,41),(5,33,17,40),(6,32,18,39),(7,25,44,24),(8,30,45,23),(9,29,46,22),(10,28,47,21),(11,27,48,20),(12,26,43,19)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,37),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 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 | 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 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | S3×D4 | S3×C4○D4 | D4○D12 |
| kernel | C6.1212+ 1+4 | S3×C22⋊C4 | Dic3⋊4D4 | D6⋊D4 | Dic3⋊D4 | C23.11D6 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | D6⋊Q8 | C2×D6⋊C4 | C23.14D6 | C3×C22.D4 | C2×C4○D12 | C2×S3×D4 | C22.D4 | C3⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 2 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C6.1212+ 1+4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
C6.1212+ 1+4 in GAP, Magma, Sage, TeX
C_6._{121}2_+^{1+4} % in TeX
G:=Group("C6.121ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1213);
// by ID
G=gap.SmallGroup(192,1213);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,346,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations