metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.49D6, C6.872+ (1+4), (C6×D4)⋊12C4, D4⋊6(C2×Dic3), (C2×D4)⋊11Dic3, (D4×Dic3)⋊37C2, (C2×D4).251D6, C6.45(C23×C4), C23⋊4(C2×Dic3), C2.5(D4⋊6D6), C12.94(C22×C4), (C2×C6).293C24, C4⋊Dic3⋊76C22, (C22×D4).13S3, (C22×C4).286D6, C2.7(C23×Dic3), (C2×C12).541C23, C3⋊4(C22.11C24), (C4×Dic3)⋊40C22, (C6×D4).270C22, C22.45(S3×C23), (C23×C6).75C22, C4.17(C22×Dic3), C6.D4⋊59C22, C23.26D6⋊32C2, (C22×C6).229C23, C23.214(C22×S3), C22.1(C22×Dic3), (C22×C12).274C22, (C2×Dic3).283C23, (C22×Dic3)⋊31C22, (D4×C2×C6).9C2, (C2×C12)⋊15(C2×C4), (C3×D4)⋊20(C2×C4), (C2×C4)⋊4(C2×Dic3), (C22×C6)⋊12(C2×C4), (C2×C6).27(C22×C4), (C2×C6.D4)⋊26C2, (C2×C4).624(C22×S3), SmallGroup(192,1357)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 712 in 338 conjugacy classes, 191 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C6, C6 [×2], C6 [×10], C2×C4 [×6], C2×C4 [×16], D4 [×16], C23, C23 [×12], C23 [×4], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×18], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×6], C3×D4 [×16], C22×C6, C22×C6 [×12], C22×C6 [×4], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C4×Dic3 [×4], C4⋊Dic3 [×4], C6.D4 [×12], C22×Dic3 [×8], C22×C12, C6×D4 [×12], C23×C6 [×2], C22.11C24, C23.26D6 [×2], D4×Dic3 [×8], C2×C6.D4 [×4], D4×C2×C6, C24.49D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C24, C2×Dic3 [×28], C22×S3 [×7], C23×C4, 2+ (1+4) [×2], C22×Dic3 [×14], S3×C23, C22.11C24, D4⋊6D6 [×2], C23×Dic3, C24.49D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
►On 48 points
(1 34)(2 32)(3 36)(4 31)(5 35)(6 33)(7 25)(8 29)(9 27)(10 37)(11 41)(12 39)(13 44)(14 48)(15 46)(16 26)(17 30)(18 28)(19 42)(20 40)(21 38)(22 43)(23 47)(24 45)
(1 7)(2 8)(3 9)(4 18)(5 16)(6 17)(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)
(1 7)(2 8)(3 9)(4 18)(5 16)(6 17)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 4)(2 5)(3 6)(7 18)(8 16)(9 17)(10 20)(11 21)(12 19)(13 23)(14 24)(15 22)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 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 22 7 10)(2 24 8 12)(3 23 9 11)(4 15 18 20)(5 14 16 19)(6 13 17 21)(25 40 34 46)(26 39 35 45)(27 38 36 44)(28 37 31 43)(29 42 32 48)(30 41 33 47)
G:=sub<Sym(48)| (1,34)(2,32)(3,36)(4,31)(5,35)(6,33)(7,25)(8,29)(9,27)(10,37)(11,41)(12,39)(13,44)(14,48)(15,46)(16,26)(17,30)(18,28)(19,42)(20,40)(21,38)(22,43)(23,47)(24,45), (1,7)(2,8)(3,9)(4,18)(5,16)(6,17)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,7)(2,8)(3,9)(4,18)(5,16)(6,17)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,4)(2,5)(3,6)(7,18)(8,16)(9,17)(10,20)(11,21)(12,19)(13,23)(14,24)(15,22)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (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), (1,22,7,10)(2,24,8,12)(3,23,9,11)(4,15,18,20)(5,14,16,19)(6,13,17,21)(25,40,34,46)(26,39,35,45)(27,38,36,44)(28,37,31,43)(29,42,32,48)(30,41,33,47)>;
G:=Group( (1,34)(2,32)(3,36)(4,31)(5,35)(6,33)(7,25)(8,29)(9,27)(10,37)(11,41)(12,39)(13,44)(14,48)(15,46)(16,26)(17,30)(18,28)(19,42)(20,40)(21,38)(22,43)(23,47)(24,45), (1,7)(2,8)(3,9)(4,18)(5,16)(6,17)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,7)(2,8)(3,9)(4,18)(5,16)(6,17)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,4)(2,5)(3,6)(7,18)(8,16)(9,17)(10,20)(11,21)(12,19)(13,23)(14,24)(15,22)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (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), (1,22,7,10)(2,24,8,12)(3,23,9,11)(4,15,18,20)(5,14,16,19)(6,13,17,21)(25,40,34,46)(26,39,35,45)(27,38,36,44)(28,37,31,43)(29,42,32,48)(30,41,33,47) );
G=PermutationGroup([(1,34),(2,32),(3,36),(4,31),(5,35),(6,33),(7,25),(8,29),(9,27),(10,37),(11,41),(12,39),(13,44),(14,48),(15,46),(16,26),(17,30),(18,28),(19,42),(20,40),(21,38),(22,43),(23,47),(24,45)], [(1,7),(2,8),(3,9),(4,18),(5,16),(6,17),(10,15),(11,13),(12,14),(19,24),(20,22),(21,23),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45)], [(1,7),(2,8),(3,9),(4,18),(5,16),(6,17),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,4),(2,5),(3,6),(7,18),(8,16),(9,17),(10,20),(11,21),(12,19),(13,23),(14,24),(15,22),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48)], [(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)], [(1,22,7,10),(2,24,8,12),(3,23,9,11),(4,15,18,20),(5,14,16,19),(6,13,17,21),(25,40,34,46),(26,39,35,45),(27,38,36,44),(28,37,31,43),(29,42,32,48),(30,41,33,47)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 6 | 7 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 6 | 7 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 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 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 8 | 10 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 10 | 9 | 0 | 0 | 0 | 0 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 5 | 4 | 11 | 2 |
| 0 | 0 | 0 | 9 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,11,0,0,0,0,0,1,0,0,0,0,0,6,0,1,0,0,0,7,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,6,0,1,0,0,0,7,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[3,11,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,4,4,0,0,0,0,8,0,3,0,0,0,10,0,0,10],[10,9,0,0,0,0,9,3,0,0,0,0,0,0,2,0,5,0,0,0,11,0,4,9,0,0,12,0,11,0,0,0,11,10,2,0] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4T | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | Dic3 | D6 | D6 | 2+ (1+4) | D4⋊6D6 |
| kernel | C24.49D6 | C23.26D6 | D4×Dic3 | C2×C6.D4 | D4×C2×C6 | C6×D4 | C22×D4 | C22×C4 | C2×D4 | C2×D4 | C24 | C6 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 16 | 1 | 1 | 8 | 4 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2^4._{49}D_6 % in TeX
G:=Group("C2^4.49D6"); // GroupNames label
G:=SmallGroup(192,1357);
// by ID
G=gap.SmallGroup(192,1357);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1123,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=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^-1>;
// generators/relations