metabelian, supersoluble, monomial
Aliases: C62.94C23, Dic32⋊19C2, C62⋊4(C2×C4), C23.19S32, C32⋊13(C4×D4), C6.162(S3×D4), C32⋊7D4⋊2C4, (C3×Dic3)⋊15D4, (C22×C6).61D6, C6.57(C4○D12), C6.D4⋊15S3, Dic3⋊8(C3⋊D4), (C2×Dic3).78D6, (C22×Dic3)⋊4S3, C3⋊6(Dic3⋊4D4), C6.D12⋊15C2, C6.44(D4⋊2S3), C62.C22⋊16C2, C2.5(D6.3D6), C22⋊2(C6.D6), (C2×C62).13C22, (C6×Dic3).71C22, (C2×C6)⋊5(C4×S3), C3⋊4(C4×C3⋊D4), C6.35(S3×C2×C4), C2.5(S3×C3⋊D4), C3⋊Dic3⋊3(C2×C4), C22.46(C2×S32), (Dic3×C2×C6)⋊14C2, C6.58(C2×C3⋊D4), (C3×C6).144(C2×D4), (C3×C6).71(C4○D4), (C2×C6.D6)⋊13C2, (C3×C6).60(C22×C4), C2.12(C2×C6.D6), (C2×C32⋊7D4).5C2, (C3×C6.D4)⋊17C2, (C2×C6).113(C22×S3), (C22×C3⋊S3).28C22, (C2×C3⋊Dic3).58C22, (C2×C3⋊S3)⋊5(C2×C4), SmallGroup(288,600)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.94C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, de=ed >
Subgroups: 802 in 215 conjugacy classes, 64 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6.D6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C62, Dic3⋊4D4, C4×C3⋊D4, Dic32, C6.D12, C62.C22, C3×C6.D4, C2×C6.D6, Dic3×C2×C6, C2×C32⋊7D4, C62.94C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, S32, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6.D6, C2×S32, Dic3⋊4D4, C4×C3⋊D4, D6.3D6, C2×C6.D6, S3×C3⋊D4, C62.94C23
►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 47 9 43 11 45)(8 48 10 44 12 46)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 42 33 38 35 40)(32 37 34 39 36 41)
(1 10 16 46)(2 11 17 47)(3 12 18 48)(4 7 13 43)(5 8 14 44)(6 9 15 45)(19 36 26 37)(20 31 27 38)(21 32 28 39)(22 33 29 40)(23 34 30 41)(24 35 25 42)
(1 31 4 34)(2 36 5 33)(3 35 6 32)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,47,9,43,11,45)(8,48,10,44,12,46)(19,28,23,26,21,30)(20,29,24,27,22,25)(31,42,33,38,35,40)(32,37,34,39,36,41), (1,10,16,46)(2,11,17,47)(3,12,18,48)(4,7,13,43)(5,8,14,44)(6,9,15,45)(19,36,26,37)(20,31,27,38)(21,32,28,39)(22,33,29,40)(23,34,30,41)(24,35,25,42), (1,31,4,34)(2,36,5,33)(3,35,6,32)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,47,9,43,11,45)(8,48,10,44,12,46)(19,28,23,26,21,30)(20,29,24,27,22,25)(31,42,33,38,35,40)(32,37,34,39,36,41), (1,10,16,46)(2,11,17,47)(3,12,18,48)(4,7,13,43)(5,8,14,44)(6,9,15,45)(19,36,26,37)(20,31,27,38)(21,32,28,39)(22,33,29,40)(23,34,30,41)(24,35,25,42), (1,31,4,34)(2,36,5,33)(3,35,6,32)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(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)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,47,9,43,11,45),(8,48,10,44,12,46),(19,28,23,26,21,30),(20,29,24,27,22,25),(31,42,33,38,35,40),(32,37,34,39,36,41)], [(1,10,16,46),(2,11,17,47),(3,12,18,48),(4,7,13,43),(5,8,14,44),(6,9,15,45),(19,36,26,37),(20,31,27,38),(21,32,28,39),(22,33,29,40),(23,34,30,41),(24,35,25,42)], [(1,31,4,34),(2,36,5,33),(3,35,6,32),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,19),(7,38),(8,39),(9,40),(10,41),(11,42),(12,37),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 6A | ··· | 6J | 6K | ··· | 6S | 12A | ··· | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
54 irreducible representations
| dim | 1 | 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 | C4 | S3 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4○D12 | S32 | S3×D4 | D4⋊2S3 | C6.D6 | C2×S32 | D6.3D6 | S3×C3⋊D4 |
| kernel | C62.94C23 | Dic32 | C6.D12 | C62.C22 | C3×C6.D4 | C2×C6.D6 | Dic3×C2×C6 | C2×C32⋊7D4 | C32⋊7D4 | C6.D4 | C22×Dic3 | C3×Dic3 | C2×Dic3 | C22×C6 | C3×C6 | Dic3 | C2×C6 | C6 | C23 | C6 | C6 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C62.94C23 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
C62.94C23 in GAP, Magma, Sage, TeX
C_6^2._{94}C_2^3 % in TeX
G:=Group("C6^2.94C2^3"); // GroupNames label
G:=SmallGroup(288,600);
// by ID
G=gap.SmallGroup(288,600);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=b^3,d^2=a^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=a^3*b^3*c,d*e=e*d>;
// generators/relations