metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.6Dic3, (C2×C6)⋊8M4(2), (C23×C6).9C4, C12.450(C2×D4), (C2×C12).484D4, (C23×C4).13S3, (C23×C12).15C2, (C22×C12).25C4, C3⋊3(C24.4C4), (C22×C4).419D6, C6.44(C2×M4(2)), C12.55D4⋊27C2, C12.89(C22⋊C4), (C2×C12).871C23, C22⋊3(C4.Dic3), (C22×C4).16Dic3, C23.33(C2×Dic3), C4.21(C6.D4), (C22×C12).543C22, C22.48(C22×Dic3), C22.18(C6.D4), (C2×C3⋊C8)⋊29C22, C4.141(C2×C3⋊D4), C6.67(C2×C22⋊C4), (C2×C4.Dic3)⋊7C2, (C2×C12).280(C2×C4), (C2×C4).64(C2×Dic3), C2.4(C2×C6.D4), C2.11(C2×C4.Dic3), (C2×C4).259(C3⋊D4), (C2×C6).191(C22×C4), (C2×C4).813(C22×S3), (C22×C6).134(C2×C4), (C2×C6).108(C22⋊C4), SmallGroup(192,766)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.6Dic3
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=de3, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 344 in 190 conjugacy classes, 79 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C22⋊C8, C2×M4(2), C23×C4, C2×C3⋊C8, C4.Dic3, C22×C12, C22×C12, C22×C12, C23×C6, C24.4C4, C12.55D4, C2×C4.Dic3, C23×C12, C24.6Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, M4(2), C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C2×M4(2), C4.Dic3, C6.D4, C22×Dic3, C2×C3⋊D4, C24.4C4, C2×C4.Dic3, C2×C6.D4, C24.6Dic3
►On 48 points
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 34)(14 35)(15 36)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)
(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 33 10 30 7 27 4 36)(2 26 11 35 8 32 5 29)(3 31 12 28 9 25 6 34)(13 42 22 39 19 48 16 45)(14 47 23 44 20 41 17 38)(15 40 24 37 21 46 18 43)
G:=sub<Sym(48)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,34)(14,35)(15,36)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,37)(11,38)(12,39)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27), (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), (1,33,10,30,7,27,4,36)(2,26,11,35,8,32,5,29)(3,31,12,28,9,25,6,34)(13,42,22,39,19,48,16,45)(14,47,23,44,20,41,17,38)(15,40,24,37,21,46,18,43)>;
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,34)(14,35)(15,36)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,37)(11,38)(12,39)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27), (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), (1,33,10,30,7,27,4,36)(2,26,11,35,8,32,5,29)(3,31,12,28,9,25,6,34)(13,42,22,39,19,48,16,45)(14,47,23,44,20,41,17,38)(15,40,24,37,21,46,18,43) );
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,34),(14,35),(15,36),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,37),(11,38),(12,39),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,25),(23,26),(24,27)], [(1,46),(2,47),(3,48),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,25),(23,26),(24,27)], [(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)], [(1,33,10,30,7,27,4,36),(2,26,11,35,8,32,5,29),(3,31,12,28,9,25,6,34),(13,42,22,39,19,48,16,45),(14,47,23,44,20,41,17,38),(15,40,24,37,21,46,18,43)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6O | 8A | ··· | 8H | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Dic3 | D6 | Dic3 | M4(2) | C3⋊D4 | C4.Dic3 |
| kernel | C24.6Dic3 | C12.55D4 | C2×C4.Dic3 | C23×C12 | C22×C12 | C23×C6 | C23×C4 | C2×C12 | C22×C4 | C22×C4 | C24 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 1 | 4 | 3 | 3 | 1 | 8 | 8 | 16 |
Matrix representation of C24.6Dic3 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,24,0,0,0,0,64,0,0,0,0,8],[0,46,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C24.6Dic3 in GAP, Magma, Sage, TeX
C_2^4._6{\rm Dic}_3 % in TeX
G:=Group("C2^4.6Dic3"); // GroupNames label
G:=SmallGroup(192,766);
// by ID
G=gap.SmallGroup(192,766);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,253,758,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,f^2=d*e^3,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*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^5>;
// generators/relations