metabelian, supersoluble, monomial
Aliases: C62.97C23, Dic32⋊21C2, C23.20S32, (C6×Dic3)⋊6C4, C62.52(C2×C4), (C2×Dic3)⋊3Dic3, C6.59(C4○D12), C62⋊5C4.1C2, (C2×Dic3).80D6, C6.D4.5S3, (C22×C6).112D6, Dic3⋊Dic3⋊18C2, C6.46(D4⋊2S3), C22.7(S3×Dic3), C32⋊9(C42⋊C2), C2.6(D6.3D6), (C2×C62).16C22, (C22×Dic3).3S3, C6.15(C22×Dic3), Dic3.10(C2×Dic3), C3⋊3(C23.26D6), C3⋊5(C23.16D6), (C6×Dic3).144C22, C6.94(S3×C2×C4), (C2×C6).75(C4×S3), C22.48(C2×S32), (Dic3×C2×C6).8C2, C2.15(C2×S3×Dic3), (C2×C6).9(C2×Dic3), (C3×C6).73(C4○D4), (C3×C6).62(C22×C4), (C2×C6).116(C22×S3), (C3×Dic3).26(C2×C4), (C3×C6.D4).5C2, (C2×C3⋊Dic3).60C22, SmallGroup(288,603)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.97C23
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=a3c, de=ed >
Subgroups: 474 in 169 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C42⋊C2, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, C62, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C62, C23.16D6, C23.26D6, Dic32, Dic3⋊Dic3, C3×C6.D4, C62⋊5C4, Dic3×C2×C6, C62.97C23
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C4×S3, C2×Dic3, C22×S3, C42⋊C2, S32, S3×C2×C4, C4○D12, D4⋊2S3, C22×Dic3, S3×Dic3, C2×S32, C23.16D6, C23.26D6, C2×S3×Dic3, D6.3D6, C62.97C23
►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 17 5 15 3 13)(2 18 6 16 4 14)(7 45 11 43 9 47)(8 46 12 44 10 48)(19 25 21 27 23 29)(20 26 22 28 24 30)(31 42 33 38 35 40)(32 37 34 39 36 41)
(1 39 15 32)(2 40 16 33)(3 41 17 34)(4 42 18 35)(5 37 13 36)(6 38 14 31)(7 27 43 19)(8 28 44 20)(9 29 45 21)(10 30 46 22)(11 25 47 23)(12 26 48 24)
(1 19 4 22)(2 24 5 21)(3 23 6 20)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 29 16 26)(14 28 17 25)(15 27 18 30)(31 44 34 47)(32 43 35 46)(33 48 36 45)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 46)(8 47)(9 48)(10 43)(11 44)(12 45)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)
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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,45,11,43,9,47)(8,46,12,44,10,48)(19,25,21,27,23,29)(20,26,22,28,24,30)(31,42,33,38,35,40)(32,37,34,39,36,41), (1,39,15,32)(2,40,16,33)(3,41,17,34)(4,42,18,35)(5,37,13,36)(6,38,14,31)(7,27,43,19)(8,28,44,20)(9,29,45,21)(10,30,46,22)(11,25,47,23)(12,26,48,24), (1,19,4,22)(2,24,5,21)(3,23,6,20)(7,42,10,39)(8,41,11,38)(9,40,12,37)(13,29,16,26)(14,28,17,25)(15,27,18,30)(31,44,34,47)(32,43,35,46)(33,48,36,45), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,46)(8,47)(9,48)(10,43)(11,44)(12,45)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40)>;
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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,45,11,43,9,47)(8,46,12,44,10,48)(19,25,21,27,23,29)(20,26,22,28,24,30)(31,42,33,38,35,40)(32,37,34,39,36,41), (1,39,15,32)(2,40,16,33)(3,41,17,34)(4,42,18,35)(5,37,13,36)(6,38,14,31)(7,27,43,19)(8,28,44,20)(9,29,45,21)(10,30,46,22)(11,25,47,23)(12,26,48,24), (1,19,4,22)(2,24,5,21)(3,23,6,20)(7,42,10,39)(8,41,11,38)(9,40,12,37)(13,29,16,26)(14,28,17,25)(15,27,18,30)(31,44,34,47)(32,43,35,46)(33,48,36,45), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,46)(8,47)(9,48)(10,43)(11,44)(12,45)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40) );
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)], [(1,17,5,15,3,13),(2,18,6,16,4,14),(7,45,11,43,9,47),(8,46,12,44,10,48),(19,25,21,27,23,29),(20,26,22,28,24,30),(31,42,33,38,35,40),(32,37,34,39,36,41)], [(1,39,15,32),(2,40,16,33),(3,41,17,34),(4,42,18,35),(5,37,13,36),(6,38,14,31),(7,27,43,19),(8,28,44,20),(9,29,45,21),(10,30,46,22),(11,25,47,23),(12,26,48,24)], [(1,19,4,22),(2,24,5,21),(3,23,6,20),(7,42,10,39),(8,41,11,38),(9,40,12,37),(13,29,16,26),(14,28,17,25),(15,27,18,30),(31,44,34,47),(32,43,35,46),(33,48,36,45)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,46),(8,47),(9,48),(10,43),(11,44),(12,45),(19,27),(20,28),(21,29),(22,30),(23,25),(24,26),(31,41),(32,42),(33,37),(34,38),(35,39),(36,40)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6J | 6K | ··· | 6S | 12A | ··· | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 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 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | Dic3 | D6 | D6 | C4○D4 | C4×S3 | C4○D12 | S32 | D4⋊2S3 | S3×Dic3 | C2×S32 | D6.3D6 |
| kernel | C62.97C23 | Dic32 | Dic3⋊Dic3 | C3×C6.D4 | C62⋊5C4 | Dic3×C2×C6 | C6×Dic3 | C6.D4 | C22×Dic3 | C2×Dic3 | C2×Dic3 | C22×C6 | C3×C6 | C2×C6 | C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 4 | 4 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 1 | 4 |
Matrix representation of C62.97C23 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 11 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,11,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,8,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
C62.97C23 in GAP, Magma, Sage, TeX
C_6^2._{97}C_2^3 % in TeX
G:=Group("C6^2.97C2^3"); // GroupNames label
G:=SmallGroup(288,603);
// by ID
G=gap.SmallGroup(288,603);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,254,1356,9414]);
// 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*c,d*e=e*d>;
// generators/relations