direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4○D12, C6.12C25, D6.6C24, D12⋊12C23, C12.47C24, C6⋊22+ (1+4), Dic6⋊14C23, Dic3.7C24, C4○D4⋊20D6, (C2×D4)⋊50D6, (C2×Q8)⋊42D6, (C4×S3)⋊2C23, (C2×C12)⋊7C23, D4⋊9(C22×S3), (C22×C4)⋊36D6, C3⋊D4⋊5C23, (C2×C6).3C24, Q8⋊9(C22×S3), (C3×Q8)⋊9C23, (C3×D4)⋊10C23, (S3×D4)⋊12C22, (C6×D4)⋊53C22, C2.13(S3×C24), C4.62(S3×C23), (C6×Q8)⋊46C22, C3⋊2(C2×2+ (1+4)), C4○D12⋊25C22, (C2×D12)⋊64C22, (C22×D12)⋊24C2, (C22×S3)⋊5C23, (S3×C23)⋊18C22, (C22×C12)⋊28C22, Q8⋊3S3⋊13C22, (C2×Dic6)⋊78C22, C22.56(S3×C23), C23.224(C22×S3), (C22×C6).248C23, (C2×Dic3).299C23, (C2×S3×D4)⋊28C2, (C2×C4○D4)⋊17S3, (C6×C4○D4)⋊14C2, (S3×C2×C4)⋊34C22, (C2×C4)⋊6(C22×S3), (C2×C4○D12)⋊37C2, (C2×Q8⋊3S3)⋊21C2, (C3×C4○D4)⋊20C22, (C2×C3⋊D4)⋊54C22, SmallGroup(192,1521)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2184 in 898 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×18], C3, C4 [×8], C4 [×4], C22, C22 [×6], C22 [×54], S3 [×12], C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×15], C2×C4 [×26], D4 [×12], D4 [×60], Q8 [×4], Q8 [×4], C23 [×3], C23 [×42], Dic3 [×4], C12 [×8], D6 [×12], D6 [×36], C2×C6, C2×C6 [×6], C2×C6 [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×D4 [×87], C2×Q8, C2×Q8, C4○D4 [×8], C4○D4 [×40], C24 [×6], Dic6 [×4], C4×S3 [×24], D12 [×36], C2×Dic3 [×2], C3⋊D4 [×24], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×S3 [×30], C22×S3 [×12], C22×C6 [×3], C22×D4 [×9], C2×C4○D4, C2×C4○D4 [×5], 2+ (1+4) [×16], C2×Dic6, S3×C2×C4 [×6], C2×D12 [×33], C4○D12 [×24], S3×D4 [×48], Q8⋊3S3 [×16], C2×C3⋊D4 [×6], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], S3×C23 [×6], C2×2+ (1+4), C22×D12 [×3], C2×C4○D12 [×3], C2×S3×D4 [×6], C2×Q8⋊3S3 [×2], D4○D12 [×16], C6×C4○D4, C2×D4○D12
Quotients:
C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], C22×S3 [×35], 2+ (1+4) [×2], C25, S3×C23 [×15], C2×2+ (1+4), D4○D12 [×2], S3×C24, C2×D4○D12
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >
►On 48 points
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 47)(14 48)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)
(1 28 7 34)(2 29 8 35)(3 30 9 36)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 44 19 38)(14 45 20 39)(15 46 21 40)(16 47 22 41)(17 48 23 42)(18 37 24 43)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 24)(10 23)(11 22)(12 21)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)
G:=sub<Sym(48)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,44,19,38)(14,45,20,39)(15,46,21,40)(16,47,22,41)(17,48,23,42)(18,37,24,43), (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,24)(10,23)(11,22)(12,21)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,44,19,38)(14,45,20,39)(15,46,21,40)(16,47,22,41)(17,48,23,42)(18,37,24,43), (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,24)(10,23)(11,22)(12,21)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43) );
G=PermutationGroup([(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,47),(14,48),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46)], [(1,28,7,34),(2,29,8,35),(3,30,9,36),(4,31,10,25),(5,32,11,26),(6,33,12,27),(13,44,19,38),(14,45,20,39),(15,46,21,40),(16,47,22,41),(17,48,23,42),(18,37,24,43)], [(1,21),(2,22),(3,23),(4,24),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,24),(10,23),(11,22),(12,21),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 3 | 6 | 3 | 6 |
| 0 | 0 | 7 | 10 | 7 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 7 | 1 |
| 0 | 0 | 6 | 3 | 12 | 6 |
| 0 | 0 | 3 | 6 | 3 | 6 |
| 0 | 0 | 7 | 10 | 7 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 2 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,3,7,0,0,7,3,6,10,0,0,0,0,3,7,0,0,0,0,6,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,3,7,0,0,7,3,6,10,0,0,7,12,3,7,0,0,1,6,6,10],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,7,3,0,0,0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,10,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,2,0,12,0,0,2,0,12,0] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | ··· | 2U | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | 2+ (1+4) | D4○D12 |
| kernel | C2×D4○D12 | C22×D12 | C2×C4○D12 | C2×S3×D4 | C2×Q8⋊3S3 | D4○D12 | C6×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 16 | 1 | 1 | 3 | 3 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_4\circ D_{12} % in TeX
G:=Group("C2xD4oD12"); // GroupNames label
G:=SmallGroup(192,1521);
// by ID
G=gap.SmallGroup(192,1521);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations