metabelian, supersoluble, monomial
Aliases: C62.19C23, Dic32⋊12C2, C12.36(C4×S3), C4⋊Dic3⋊11S3, (C2×C12).129D6, C6.2(D4⋊2S3), (C6×C12).89C22, (C2×Dic3).56D6, C2.2(D12⋊S3), C6.22(Q8⋊3S3), C32⋊4(C42⋊C2), C6.D12.6C2, C4.10(C6.D6), (C6×Dic3).54C22, (C4×C3⋊S3)⋊1C4, C6.30(S3×C2×C4), (C2×C4).112S32, (C3×C4⋊Dic3)⋊6C2, C22.22(C2×S32), C3⋊2(C4⋊C4⋊7S3), (C3×C12).64(C2×C4), (C3×C6).8(C4○D4), C2.8(C2×C6.D6), C3⋊Dic3.43(C2×C4), (C2×C6).38(C22×S3), (C3×C6).50(C22×C4), (C22×C3⋊S3).63C22, (C2×C3⋊Dic3).115C22, (C2×C4×C3⋊S3).2C2, (C2×C3⋊S3).37(C2×C4), SmallGroup(288,497)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.19C23
G = < a,b,c,d,e | a6=b6=1, c2=a3, d2=a3b3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >
Subgroups: 658 in 179 conjugacy classes, 60 normal (12 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊C2, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4⋊C4⋊7S3, Dic32, C6.D12, C3×C4⋊Dic3, C2×C4×C3⋊S3, C62.19C23
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, S32, S3×C2×C4, D4⋊2S3, Q8⋊3S3, C6.D6, C2×S32, C4⋊C4⋊7S3, D12⋊S3, C2×C6.D6, C62.19C23
►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 18 5 16 3 14)(2 13 6 17 4 15)(7 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 45 4 48)(2 46 5 43)(3 47 6 44)(7 13 10 16)(8 14 11 17)(9 15 12 18)(19 39 22 42)(20 40 23 37)(21 41 24 38)(25 31 28 34)(26 32 29 35)(27 33 30 36)
(1 38 13 36)(2 37 14 35)(3 42 15 34)(4 41 16 33)(5 40 17 32)(6 39 18 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 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 33 45 38)(8 34 46 39)(9 35 47 40)(10 36 48 41)(11 31 43 42)(12 32 44 37)
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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,43,9,45,11,47)(8,44,10,46,12,48)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39), (1,45,4,48)(2,46,5,43)(3,47,6,44)(7,13,10,16)(8,14,11,17)(9,15,12,18)(19,39,22,42)(20,40,23,37)(21,41,24,38)(25,31,28,34)(26,32,29,35)(27,33,30,36), (1,38,13,36)(2,37,14,35)(3,42,15,34)(4,41,16,33)(5,40,17,32)(6,39,18,31)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19), (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,33,45,38)(8,34,46,39)(9,35,47,40)(10,36,48,41)(11,31,43,42)(12,32,44,37)>;
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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,43,9,45,11,47)(8,44,10,46,12,48)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39), (1,45,4,48)(2,46,5,43)(3,47,6,44)(7,13,10,16)(8,14,11,17)(9,15,12,18)(19,39,22,42)(20,40,23,37)(21,41,24,38)(25,31,28,34)(26,32,29,35)(27,33,30,36), (1,38,13,36)(2,37,14,35)(3,42,15,34)(4,41,16,33)(5,40,17,32)(6,39,18,31)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19), (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,33,45,38)(8,34,46,39)(9,35,47,40)(10,36,48,41)(11,31,43,42)(12,32,44,37) );
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)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,43,9,45,11,47),(8,44,10,46,12,48),(19,30,23,28,21,26),(20,25,24,29,22,27),(31,40,33,42,35,38),(32,41,34,37,36,39)], [(1,45,4,48),(2,46,5,43),(3,47,6,44),(7,13,10,16),(8,14,11,17),(9,15,12,18),(19,39,22,42),(20,40,23,37),(21,41,24,38),(25,31,28,34),(26,32,29,35),(27,33,30,36)], [(1,38,13,36),(2,37,14,35),(3,42,15,34),(4,41,16,33),(5,40,17,32),(6,39,18,31),(7,30,48,24),(8,29,43,23),(9,28,44,22),(10,27,45,21),(11,26,46,20),(12,25,47,19)], [(1,27,16,24),(2,28,17,19),(3,29,18,20),(4,30,13,21),(5,25,14,22),(6,26,15,23),(7,33,45,38),(8,34,46,39),(9,35,47,40),(10,36,48,41),(11,31,43,42),(12,32,44,37)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | ··· | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 12A | ··· | 12H | 12I | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 6 | ··· | 6 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | S32 | D4⋊2S3 | Q8⋊3S3 | C6.D6 | C2×S32 | D12⋊S3 |
| kernel | C62.19C23 | Dic32 | C6.D12 | C3×C4⋊Dic3 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C4⋊Dic3 | C2×Dic3 | C2×C12 | C3×C6 | C12 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 2 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 2 | 1 | 4 |
Matrix representation of C62.19C23 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,3,5,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
C62.19C23 in GAP, Magma, Sage, TeX
C_6^2._{19}C_2^3 % in TeX
G:=Group("C6^2.19C2^3"); // GroupNames label
G:=SmallGroup(288,497);
// by ID
G=gap.SmallGroup(288,497);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,176,422,219,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=a^3,d^2=a^3*b^3,e^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
// generators/relations