metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊12D6, C6.892+ (1+4), (C2×D4)⋊40D6, (C22×C4)⋊31D6, (C22×C6)⋊13D4, C23⋊2D6⋊30C2, D6⋊C4⋊36C22, (C22×D4)⋊13S3, (C6×D4)⋊58C22, C3⋊5(C23⋊3D4), C23⋊5(C3⋊D4), C24⋊4S3⋊12C2, (C2×C6).299C24, (C23×C6)⋊14C22, C6.146(C22×D4), C23.14D6⋊41C2, C2.92(D4⋊6D6), (C2×C12).644C23, Dic3⋊C4⋊38C22, (S3×C23)⋊15C22, (C22×C12)⋊44C22, C6.D4⋊64C22, C23.28D6⋊28C2, C23.23D6⋊29C2, (C22×C6).233C23, C23.216(C22×S3), C22.312(S3×C23), (C22×S3).130C23, (C2×Dic3).154C23, (C22×Dic3)⋊34C22, (D4×C2×C6)⋊17C2, (C2×C6).582(C2×D4), (C2×C3⋊D4)⋊48C22, (C22×C3⋊D4)⋊17C2, C22.20(C2×C3⋊D4), C2.19(C22×C3⋊D4), (C2×C6.D4)⋊30C2, (C2×C4).238(C22×S3), SmallGroup(192,1363)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 968 in 346 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×6], C22 [×30], S3 [×2], C6, C6 [×2], C6 [×8], C2×C4 [×2], C2×C4 [×12], D4 [×20], C23 [×3], C23 [×6], C23 [×12], Dic3 [×6], C12 [×2], D6 [×10], C2×C6, C2×C6 [×6], C2×C6 [×20], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×4], C2×D4 [×16], C24 [×2], C24, C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×S3 [×4], C22×C6 [×3], C22×C6 [×6], C22×C6 [×6], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4, C22×D4, Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×8], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], S3×C23, C23×C6 [×2], C23⋊3D4, C23.28D6 [×2], C23.23D6 [×2], C23⋊2D6 [×2], C23.14D6 [×4], C2×C6.D4, C24⋊4S3 [×2], C22×C3⋊D4, D4×C2×C6, C24⋊12D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, 2+ (1+4) [×2], C2×C3⋊D4 [×6], S3×C23, C23⋊3D4, D4⋊6D6 [×2], C22×C3⋊D4, C24⋊12D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
►On 48 points
(1 15)(2 13)(3 14)(4 18)(5 16)(6 17)(7 23)(8 24)(9 22)(10 20)(11 21)(12 19)(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 35)(2 33)(3 31)(4 27)(5 25)(6 29)(7 43)(8 47)(9 45)(10 46)(11 44)(12 48)(13 30)(14 28)(15 26)(16 34)(17 32)(18 36)(19 42)(20 40)(21 38)(22 39)(23 37)(24 41)
(1 22)(2 23)(3 24)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)(16 21)(17 19)(18 20)(25 44)(26 45)(27 46)(28 47)(29 48)(30 43)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(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 18)(2 17)(3 16)(4 6)(7 9)(10 12)(13 15)(19 23)(20 22)(21 24)(25 47)(26 46)(27 45)(28 44)(29 43)(30 48)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)
G:=sub<Sym(48)| (1,15)(2,13)(3,14)(4,18)(5,16)(6,17)(7,23)(8,24)(9,22)(10,20)(11,21)(12,19)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,35)(2,33)(3,31)(4,27)(5,25)(6,29)(7,43)(8,47)(9,45)(10,46)(11,44)(12,48)(13,30)(14,28)(15,26)(16,34)(17,32)(18,36)(19,42)(20,40)(21,38)(22,39)(23,37)(24,41), (1,22)(2,23)(3,24)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15)(16,21)(17,19)(18,20)(25,44)(26,45)(27,46)(28,47)(29,48)(30,43)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40), (1,17)(2,18)(3,16)(4,13)(5,14)(6,15)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(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), (1,18)(2,17)(3,16)(4,6)(7,9)(10,12)(13,15)(19,23)(20,22)(21,24)(25,47)(26,46)(27,45)(28,44)(29,43)(30,48)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42)>;
G:=Group( (1,15)(2,13)(3,14)(4,18)(5,16)(6,17)(7,23)(8,24)(9,22)(10,20)(11,21)(12,19)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,35)(2,33)(3,31)(4,27)(5,25)(6,29)(7,43)(8,47)(9,45)(10,46)(11,44)(12,48)(13,30)(14,28)(15,26)(16,34)(17,32)(18,36)(19,42)(20,40)(21,38)(22,39)(23,37)(24,41), (1,22)(2,23)(3,24)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15)(16,21)(17,19)(18,20)(25,44)(26,45)(27,46)(28,47)(29,48)(30,43)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40), (1,17)(2,18)(3,16)(4,13)(5,14)(6,15)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(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), (1,18)(2,17)(3,16)(4,6)(7,9)(10,12)(13,15)(19,23)(20,22)(21,24)(25,47)(26,46)(27,45)(28,44)(29,43)(30,48)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42) );
G=PermutationGroup([(1,15),(2,13),(3,14),(4,18),(5,16),(6,17),(7,23),(8,24),(9,22),(10,20),(11,21),(12,19),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,35),(2,33),(3,31),(4,27),(5,25),(6,29),(7,43),(8,47),(9,45),(10,46),(11,44),(12,48),(13,30),(14,28),(15,26),(16,34),(17,32),(18,36),(19,42),(20,40),(21,38),(22,39),(23,37),(24,41)], [(1,22),(2,23),(3,24),(4,10),(5,11),(6,12),(7,13),(8,14),(9,15),(16,21),(17,19),(18,20),(25,44),(26,45),(27,46),(28,47),(29,48),(30,43),(31,41),(32,42),(33,37),(34,38),(35,39),(36,40)], [(1,17),(2,18),(3,16),(4,13),(5,14),(6,15),(7,10),(8,11),(9,12),(19,22),(20,23),(21,24),(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)], [(1,18),(2,17),(3,16),(4,6),(7,9),(10,12),(13,15),(19,23),(20,22),(21,24),(25,47),(26,46),(27,45),(28,44),(29,43),(30,48),(31,41),(32,40),(33,39),(34,38),(35,37),(36,42)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 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 | 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 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,0,12,0,0,0,0,11,0,12],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,12,0,0,0,0,0,0,12] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | ··· | 4H | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 12 | 12 | 2 | 4 | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C3⋊D4 | 2+ (1+4) | D4⋊6D6 |
| kernel | C24⋊12D6 | C23.28D6 | C23.23D6 | C23⋊2D6 | C23.14D6 | C2×C6.D4 | C24⋊4S3 | C22×C3⋊D4 | D4×C2×C6 | C22×D4 | C22×C6 | C22×C4 | C2×D4 | C24 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes_{12}D_6 % in TeX
G:=Group("C2^4:12D6"); // GroupNames label
G:=SmallGroup(192,1363);
// by ID
G=gap.SmallGroup(192,1363);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations