metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.53D6, C6.912+ 1+4, (C3×D4)⋊17D4, D4⋊8(C3⋊D4), D6⋊3D4⋊41C2, C3⋊10(D4⋊5D4), D6⋊C4⋊73C22, (D4×Dic3)⋊39C2, (C2×D4).231D6, C12.253(C2×D4), (C22×D4)⋊15S3, C24⋊4S3⋊14C2, (C2×C6).301C24, C4⋊Dic3⋊45C22, (C22×C4).289D6, C6.148(C22×D4), C23.14D6⋊42C2, C23.12D6⋊29C2, C12.48D4⋊37C2, C2.94(D4⋊6D6), C22⋊6(D4⋊2S3), (C2×C12).546C23, Dic3⋊C4⋊39C22, (C2×Dic6)⋊42C22, (C4×Dic3)⋊43C22, (C6×D4).312C22, (C23×C6).80C22, C23.23D6⋊30C2, C6.D4⋊40C22, C22.314(S3×C23), C23.246(C22×S3), (C22×C6).235C23, (C22×S3).132C23, (C22×C12).278C22, (C2×Dic3).286C23, (C22×Dic3)⋊35C22, (D4×C2×C6)⋊8C2, (C4×C3⋊D4)⋊26C2, (S3×C2×C4)⋊32C22, (C2×C6).74(C2×D4), C4.68(C2×C3⋊D4), (C2×C6)⋊15(C4○D4), C6.107(C2×C4○D4), (C2×D4⋊2S3)⋊27C2, C22.3(C2×C3⋊D4), C2.71(C2×D4⋊2S3), (C2×C3⋊D4)⋊29C22, C2.21(C22×C3⋊D4), (C2×C6.D4)⋊31C2, (C2×C4).239(C22×S3), SmallGroup(192,1365)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.53D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 824 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C6×D4, C23×C6, D4⋊5D4, C12.48D4, C4×C3⋊D4, D4×Dic3, C23.23D6, C23.12D6, D6⋊3D4, C23.14D6, C2×C6.D4, C24⋊4S3, C2×D4⋊2S3, D4×C2×C6, C24.53D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊2S3, C2×C3⋊D4, S3×C23, D4⋊5D4, C2×D4⋊2S3, D4⋊6D6, C22×C3⋊D4, C24.53D6
►On 48 points
(1 33)(2 28)(3 35)(4 30)(5 25)(6 32)(7 27)(8 34)(9 29)(10 36)(11 31)(12 26)(13 42)(14 37)(15 44)(16 39)(17 46)(18 41)(19 48)(20 43)(21 38)(22 45)(23 40)(24 47)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 37)(21 38)(22 39)(23 40)(24 41)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(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 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 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 14 7 20)(2 19 8 13)(3 24 9 18)(4 17 10 23)(5 22 11 16)(6 15 12 21)(25 45 31 39)(26 38 32 44)(27 43 33 37)(28 48 34 42)(29 41 35 47)(30 46 36 40)
G:=sub<Sym(48)| (1,33)(2,28)(3,35)(4,30)(5,25)(6,32)(7,27)(8,34)(9,29)(10,36)(11,31)(12,26)(13,42)(14,37)(15,44)(16,39)(17,46)(18,41)(19,48)(20,43)(21,38)(22,45)(23,40)(24,47), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,48)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47), (1,7)(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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,14,7,20)(2,19,8,13)(3,24,9,18)(4,17,10,23)(5,22,11,16)(6,15,12,21)(25,45,31,39)(26,38,32,44)(27,43,33,37)(28,48,34,42)(29,41,35,47)(30,46,36,40)>;
G:=Group( (1,33)(2,28)(3,35)(4,30)(5,25)(6,32)(7,27)(8,34)(9,29)(10,36)(11,31)(12,26)(13,42)(14,37)(15,44)(16,39)(17,46)(18,41)(19,48)(20,43)(21,38)(22,45)(23,40)(24,47), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,48)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47), (1,7)(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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,14,7,20)(2,19,8,13)(3,24,9,18)(4,17,10,23)(5,22,11,16)(6,15,12,21)(25,45,31,39)(26,38,32,44)(27,43,33,37)(28,48,34,42)(29,41,35,47)(30,46,36,40) );
G=PermutationGroup([[(1,33),(2,28),(3,35),(4,30),(5,25),(6,32),(7,27),(8,34),(9,29),(10,36),(11,31),(12,26),(13,42),(14,37),(15,44),(16,39),(17,46),(18,41),(19,48),(20,43),(21,38),(22,45),(23,40),(24,47)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,37),(21,38),(22,39),(23,40),(24,41),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,33),(2,34),(3,35),(4,36),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,48),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47)], [(1,7),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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)], [(1,14,7,20),(2,19,8,13),(3,24,9,18),(4,17,10,23),(5,22,11,16),(6,15,12,21),(25,45,31,39),(26,38,32,44),(27,43,33,37),(28,48,34,42),(29,41,35,47),(30,46,36,40)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 12 | 2 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | 2+ 1+4 | D4⋊2S3 | D4⋊6D6 |
| kernel | C24.53D6 | C12.48D4 | C4×C3⋊D4 | D4×Dic3 | C23.23D6 | C23.12D6 | D6⋊3D4 | C23.14D6 | C2×C6.D4 | C24⋊4S3 | C2×D4⋊2S3 | D4×C2×C6 | C22×D4 | C3×D4 | C22×C4 | C2×D4 | C24 | C2×C6 | D4 | C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C24.53D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 10 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 11 | 12 |
| 0 | 9 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[10,0,0,0,0,4,0,0,0,0,1,11,0,0,1,12],[0,3,0,0,9,0,0,0,0,0,8,0,0,0,0,8] >;
C24.53D6 in GAP, Magma, Sage, TeX
C_2^4._{53}D_6 % in TeX
G:=Group("C2^4.53D6"); // GroupNames label
G:=SmallGroup(192,1365);
// by ID
G=gap.SmallGroup(192,1365);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,675,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*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=e^5>;
// generators/relations