metabelian, supersoluble, monomial
Aliases: C62.20C23, Dic3⋊C4⋊1S3, (S3×C6).28D4, C6.134(S3×D4), D6⋊Dic3⋊29C2, (C2×C12).257D6, (C2×Dic3).7D6, C6.18(C4○D12), C3⋊6(D6.D4), D6.11(C3⋊D4), C6.5(Q8⋊3S3), C62.C22⋊2C2, (C22×S3).59D6, C6.D12⋊20C2, C6.11D12⋊13C2, (C6×C12).211C22, C2.6(D6.D6), C2.8(D6.6D6), C3⋊1(C23.28D6), C32⋊2(C22.D4), (C6×Dic3).106C22, (S3×C2×C4)⋊7S3, (C2×C4).39S32, (S3×C2×C12)⋊15C2, C2.9(S3×C3⋊D4), C22.78(C2×S32), (C3×C6).78(C2×D4), C6.27(C2×C3⋊D4), (C3×Dic3⋊C4)⋊3C2, (C3×C6).9(C4○D4), (S3×C2×C6).69C22, (C2×C3⋊D12).1C2, (C2×C6).39(C22×S3), (C22×C3⋊S3).8C22, (C2×C3⋊Dic3).20C22, SmallGroup(288,498)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.20C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a3c, ce=ec, ede-1=b3d >
Subgroups: 730 in 173 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, Dic3⋊C4, Dic3⋊C4, D6⋊C4, C6.D4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C3⋊D12, S3×C12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, D6.D4, C23.28D6, D6⋊Dic3, C6.D12, C62.C22, C3×Dic3⋊C4, C6.11D12, C2×C3⋊D12, S3×C2×C12, C62.20C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, S32, C4○D12, S3×D4, Q8⋊3S3, C2×C3⋊D4, C2×S32, D6.D4, C23.28D6, D6.D6, D6.6D6, S3×C3⋊D4, C62.20C23
►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 15 3 17 5 13)(2 16 4 18 6 14)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 37 35 41 33 39)(32 38 36 42 34 40)
(1 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(19 38)(20 39)(21 40)(22 41)(23 42)(24 37)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)
(1 40 17 36)(2 39 18 35)(3 38 13 34)(4 37 14 33)(5 42 15 32)(6 41 16 31)(7 30 48 24)(8 29 43 23)(9 28 44 22)(10 27 45 21)(11 26 46 20)(12 25 47 19)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 36 10 33)(8 31 11 34)(9 32 12 35)(13 29 16 26)(14 30 17 27)(15 25 18 28)(37 48 40 45)(38 43 41 46)(39 44 42 47)
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), (1,15,3,17,5,13)(2,16,4,18,6,14)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,29,21,25,23,27)(20,30,22,26,24,28)(31,37,35,41,33,39)(32,38,36,42,34,40), (1,48)(2,43)(3,44)(4,45)(5,46)(6,47)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,38)(20,39)(21,40)(22,41)(23,42)(24,37)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33), (1,40,17,36)(2,39,18,35)(3,38,13,34)(4,37,14,33)(5,42,15,32)(6,41,16,31)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,36,10,33)(8,31,11,34)(9,32,12,35)(13,29,16,26)(14,30,17,27)(15,25,18,28)(37,48,40,45)(38,43,41,46)(39,44,42,47)>;
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), (1,15,3,17,5,13)(2,16,4,18,6,14)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,29,21,25,23,27)(20,30,22,26,24,28)(31,37,35,41,33,39)(32,38,36,42,34,40), (1,48)(2,43)(3,44)(4,45)(5,46)(6,47)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,38)(20,39)(21,40)(22,41)(23,42)(24,37)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33), (1,40,17,36)(2,39,18,35)(3,38,13,34)(4,37,14,33)(5,42,15,32)(6,41,16,31)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,36,10,33)(8,31,11,34)(9,32,12,35)(13,29,16,26)(14,30,17,27)(15,25,18,28)(37,48,40,45)(38,43,41,46)(39,44,42,47) );
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)], [(1,15,3,17,5,13),(2,16,4,18,6,14),(7,44,11,48,9,46),(8,45,12,43,10,47),(19,29,21,25,23,27),(20,30,22,26,24,28),(31,37,35,41,33,39),(32,38,36,42,34,40)], [(1,48),(2,43),(3,44),(4,45),(5,46),(6,47),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16),(19,38),(20,39),(21,40),(22,41),(23,42),(24,37),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33)], [(1,40,17,36),(2,39,18,35),(3,38,13,34),(4,37,14,33),(5,42,15,32),(6,41,16,31),(7,30,48,24),(8,29,43,23),(9,28,44,22),(10,27,45,21),(11,26,46,20),(12,25,47,19)], [(1,21,4,24),(2,22,5,19),(3,23,6,20),(7,36,10,33),(8,31,11,34),(9,32,12,35),(13,29,16,26),(14,30,17,27),(15,25,18,28),(37,48,40,45),(38,43,41,46),(39,44,42,47)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 12Q | 12R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 36 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 12 | 12 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D12 | S32 | S3×D4 | Q8⋊3S3 | C2×S32 | D6.D6 | D6.6D6 | S3×C3⋊D4 |
| kernel | C62.20C23 | D6⋊Dic3 | C6.D12 | C62.C22 | C3×Dic3⋊C4 | C6.11D12 | C2×C3⋊D12 | S3×C2×C12 | Dic3⋊C4 | S3×C2×C4 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | D6 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 1 | 4 | 4 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C62.20C23 ►in GL6(𝔽13)
| 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 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 6 | 3 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[6,5,0,0,0,0,3,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,3,10],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
C62.20C23 in GAP, Magma, Sage, TeX
C_6^2._{20}C_2^3 % in TeX
G:=Group("C6^2.20C2^3"); // GroupNames label
G:=SmallGroup(288,498);
// by ID
G=gap.SmallGroup(288,498);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=b^3,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=b^3*d>;
// generators/relations