metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.59D6, C23.49D12, (C22×C4)⋊4D6, (S3×C23)⋊5C4, C3⋊1(C24⋊3C4), D6⋊3(C22⋊C4), C6.37C22≀C2, C22⋊4(D6⋊C4), (S3×C24).1C2, C23.56(C4×S3), (C22×C6).67D4, C2.2(C23⋊2D6), C2.4(D6⋊D4), (C22×C12)⋊1C22, (C22×S3).87D4, C22.100(S3×D4), C22.43(C2×D12), C23.59(C3⋊D4), (C23×C6).38C22, (S3×C23).87C22, C23.292(C22×S3), (C22×C6).329C23, (C22×Dic3)⋊2C22, (C2×D6⋊C4)⋊3C2, C2.9(C2×D6⋊C4), (C2×C22⋊C4)⋊2S3, (C6×C22⋊C4)⋊2C2, (C2×C6)⋊1(C22⋊C4), (C2×C6).321(C2×D4), C2.28(S3×C22⋊C4), C6.36(C2×C22⋊C4), C22.126(S3×C2×C4), (C2×C6.D4)⋊2C2, (C22×C6).53(C2×C4), C22.50(C2×C3⋊D4), (C22×S3).60(C2×C4), (C2×C6).108(C22×C4), SmallGroup(192,514)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.59D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=cb=bc, ab=ba, ac=ca, eae-1=faf-1=ad=da, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >
Subgroups: 1608 in 506 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C24, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C25, D6⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, S3×C23, S3×C23, C23×C6, C24⋊3C4, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, S3×C24, C24.59D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C22≀C2, D6⋊C4, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C24⋊3C4, S3×C22⋊C4, D6⋊D4, C2×D6⋊C4, C23⋊2D6, C24.59D6
►On 48 points
(1 14)(2 38)(3 16)(4 40)(5 18)(6 42)(7 20)(8 44)(9 22)(10 46)(11 24)(12 48)(13 30)(15 32)(17 34)(19 36)(21 26)(23 28)(25 43)(27 45)(29 47)(31 37)(33 39)(35 41)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 48)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)
(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 36 43 13)(2 24 44 35)(3 34 45 23)(4 22 46 33)(5 32 47 21)(6 20 48 31)(7 30 37 19)(8 18 38 29)(9 28 39 17)(10 16 40 27)(11 26 41 15)(12 14 42 25)
G:=sub<Sym(48)| (1,14)(2,38)(3,16)(4,40)(5,18)(6,42)(7,20)(8,44)(9,22)(10,46)(11,24)(12,48)(13,30)(15,32)(17,34)(19,36)(21,26)(23,28)(25,43)(27,45)(29,47)(31,37)(33,39)(35,41), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,48)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47), (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), (1,36,43,13)(2,24,44,35)(3,34,45,23)(4,22,46,33)(5,32,47,21)(6,20,48,31)(7,30,37,19)(8,18,38,29)(9,28,39,17)(10,16,40,27)(11,26,41,15)(12,14,42,25)>;
G:=Group( (1,14)(2,38)(3,16)(4,40)(5,18)(6,42)(7,20)(8,44)(9,22)(10,46)(11,24)(12,48)(13,30)(15,32)(17,34)(19,36)(21,26)(23,28)(25,43)(27,45)(29,47)(31,37)(33,39)(35,41), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,48)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47), (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), (1,36,43,13)(2,24,44,35)(3,34,45,23)(4,22,46,33)(5,32,47,21)(6,20,48,31)(7,30,37,19)(8,18,38,29)(9,28,39,17)(10,16,40,27)(11,26,41,15)(12,14,42,25) );
G=PermutationGroup([[(1,14),(2,38),(3,16),(4,40),(5,18),(6,42),(7,20),(8,44),(9,22),(10,46),(11,24),(12,48),(13,30),(15,32),(17,34),(19,36),(21,26),(23,28),(25,43),(27,45),(29,47),(31,37),(33,39),(35,41)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,25),(21,26),(22,27),(23,28),(24,29)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,48),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47)], [(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)], [(1,36,43,13),(2,24,44,35),(3,34,45,23),(4,22,46,33),(5,32,47,21),(6,20,48,31),(7,30,37,19),(8,18,38,29),(9,28,39,17),(10,16,40,27),(11,26,41,15),(12,14,42,25)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | S3×D4 |
| kernel | C24.59D6 | C2×D6⋊C4 | C2×C6.D4 | C6×C22⋊C4 | S3×C24 | S3×C23 | C2×C22⋊C4 | C22×S3 | C22×C6 | C22×C4 | C24 | C23 | C23 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 8 | 4 | 2 | 1 | 4 | 4 | 4 | 4 |
Matrix representation of C24.59D6 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 9 | 2 | 0 | 0 |
| 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 2 | 9 | 0 | 0 |
| 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,0,9,11,0,0,0,2,11,0,0,0,0,0,0,12,0,0,0,1,0],[8,0,0,0,0,0,2,11,0,0,0,9,11,0,0,0,0,0,0,1,0,0,0,1,0] >;
C24.59D6 in GAP, Magma, Sage, TeX
C_2^4._{59}D_6 % in TeX
G:=Group("C2^4.59D6"); // GroupNames label
G:=SmallGroup(192,514);
// by ID
G=gap.SmallGroup(192,514);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;
// generators/relations