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, C2.94(D4⋊6D6), C12.48D4⋊37C2, C22⋊6(D4⋊2S3), (C2×C12).546C23, Dic3⋊C4⋊39C22, (C4×Dic3)⋊43C22, (C2×Dic6)⋊42C22, (C6×D4).312C22, (C23×C6).80C22, C23.23D6⋊30C2, C6.D4⋊40C22, C22.314(S3×C23), (C22×C6).235C23, C23.246(C22×S3), (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
Subgroups: 824 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], S3, C6 [×3], C6 [×8], C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×11], Dic3 [×7], C12 [×2], C12, D6 [×3], C2×C6, C2×C6 [×6], C2×C6 [×20], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C22×S3, C22×C6, C22×C6 [×4], C22×C6 [×10], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4, C6.D4, C6.D4 [×10], C2×Dic6, S3×C2×C4, D4⋊2S3 [×4], C22×Dic3 [×4], C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C6×D4 [×2], C6×D4 [×2], C6×D4 [×4], C23×C6 [×2], D4⋊5D4, C12.48D4, C4×C3⋊D4, D4×Dic3, C23.23D6 [×2], C23.12D6, D6⋊3D4, C23.14D6 [×2], C2×C6.D4 [×2], C24⋊4S3 [×2], C2×D4⋊2S3, D4×C2×C6, C24.53D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), D4⋊2S3 [×2], C2×C3⋊D4 [×6], S3×C23, D4⋊5D4, C2×D4⋊2S3, D4⋊6D6, C22×C3⋊D4, C24.53D6
Generators and relations
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 >
►On 48 points
(1 20)(2 15)(3 22)(4 17)(5 24)(6 19)(7 14)(8 21)(9 16)(10 23)(11 18)(12 13)(25 40)(26 47)(27 42)(28 37)(29 44)(30 39)(31 46)(32 41)(33 48)(34 43)(35 38)(36 45)
(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 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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 48 7 42)(2 41 8 47)(3 46 9 40)(4 39 10 45)(5 44 11 38)(6 37 12 43)(13 34 19 28)(14 27 20 33)(15 32 21 26)(16 25 22 31)(17 30 23 36)(18 35 24 29)
G:=sub<Sym(48)| (1,20)(2,15)(3,22)(4,17)(5,24)(6,19)(7,14)(8,21)(9,16)(10,23)(11,18)(12,13)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (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), (1,48,7,42)(2,41,8,47)(3,46,9,40)(4,39,10,45)(5,44,11,38)(6,37,12,43)(13,34,19,28)(14,27,20,33)(15,32,21,26)(16,25,22,31)(17,30,23,36)(18,35,24,29)>;
G:=Group( (1,20)(2,15)(3,22)(4,17)(5,24)(6,19)(7,14)(8,21)(9,16)(10,23)(11,18)(12,13)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (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), (1,48,7,42)(2,41,8,47)(3,46,9,40)(4,39,10,45)(5,44,11,38)(6,37,12,43)(13,34,19,28)(14,27,20,33)(15,32,21,26)(16,25,22,31)(17,30,23,36)(18,35,24,29) );
G=PermutationGroup([(1,20),(2,15),(3,22),(4,17),(5,24),(6,19),(7,14),(8,21),(9,16),(10,23),(11,18),(12,13),(25,40),(26,47),(27,42),(28,37),(29,44),(30,39),(31,46),(32,41),(33,48),(34,43),(35,38),(36,45)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,46),(26,47),(27,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,37),(35,38),(36,39)], [(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)], [(1,48,7,42),(2,41,8,47),(3,46,9,40),(4,39,10,45),(5,44,11,38),(6,37,12,43),(13,34,19,28),(14,27,20,33),(15,32,21,26),(16,25,22,31),(17,30,23,36),(18,35,24,29)])
Matrix representation ►G ⊆ 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] >;
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 |
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