direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊6D6, C24⋊13D6, C6.7C25, D12⋊9C23, D6.3C24, C12.42C24, Dic6⋊9C23, C6⋊12+ (1+4), Dic3.3C24, (C2×D4)⋊46D6, (C4×S3)⋊1C23, (C2×C12)⋊5C23, (C3×D4)⋊8C23, D4⋊7(C22×S3), (C22×C4)⋊35D6, C3⋊D4⋊3C23, C2.8(S3×C24), (C22×D4)⋊16S3, (S3×D4)⋊11C22, (C6×D4)⋊51C22, C4.42(S3×C23), (C22×C6)⋊7C23, C23⋊5(C22×S3), C3⋊1(C2×2+ (1+4)), C4○D12⋊22C22, (C2×D12)⋊61C22, (C2×C6).327C24, (C22×S3)⋊4C23, (C23×C6)⋊16C22, (C2×Dic3)⋊5C23, C22.8(S3×C23), D4⋊2S3⋊12C22, (S3×C23)⋊17C22, (C22×C12)⋊26C22, (C2×Dic6)⋊72C22, (C22×Dic3)⋊38C22, (D4×C2×C6)⋊11C2, (C2×S3×D4)⋊27C2, (S3×C2×C4)⋊33C22, (C2×C4)⋊5(C22×S3), (C2×C4○D12)⋊34C2, (C2×D4⋊2S3)⋊29C2, (C22×C3⋊D4)⋊21C2, (C2×C3⋊D4)⋊52C22, SmallGroup(192,1516)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2056 in 898 conjugacy classes, 447 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×18], C3, C4 [×4], C4 [×8], C22, C22 [×10], C22 [×50], S3 [×8], C6, C6 [×2], C6 [×10], C2×C4 [×6], C2×C4 [×36], D4 [×16], D4 [×56], Q8 [×8], C23, C23 [×12], C23 [×32], Dic3 [×8], C12 [×4], D6 [×8], D6 [×24], C2×C6, C2×C6 [×10], C2×C6 [×18], C22×C4, C22×C4 [×8], C2×D4 [×12], C2×D4 [×78], C2×Q8 [×2], C4○D4 [×48], C24 [×2], C24 [×4], Dic6 [×8], C4×S3 [×16], D12 [×8], C2×Dic3 [×20], C3⋊D4 [×48], C2×C12 [×6], C3×D4 [×16], C22×S3 [×20], C22×S3 [×8], C22×C6, C22×C6 [×12], C22×C6 [×4], C22×D4, C22×D4 [×8], C2×C4○D4 [×6], 2+ (1+4) [×16], C2×Dic6 [×2], S3×C2×C4 [×4], C2×D12 [×2], C4○D12 [×16], S3×D4 [×32], D4⋊2S3 [×32], C22×Dic3 [×4], C2×C3⋊D4 [×44], C22×C12, C6×D4 [×12], S3×C23 [×4], C23×C6 [×2], C2×2+ (1+4), C2×C4○D12 [×2], C2×S3×D4 [×4], C2×D4⋊2S3 [×4], D4⋊6D6 [×16], C22×C3⋊D4 [×4], D4×C2×C6, C2×D4⋊6D6
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⋊6D6 [×2], S3×C24, C2×D4⋊6D6
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
►On 48 points
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(13 27)(14 28)(15 29)(16 30)(17 25)(18 26)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 31 23 44)(2 45 24 32)(3 33 19 46)(4 47 20 34)(5 35 21 48)(6 43 22 36)(7 18 29 42)(8 37 30 13)(9 14 25 38)(10 39 26 15)(11 16 27 40)(12 41 28 17)
(1 38)(2 15)(3 40)(4 17)(5 42)(6 13)(7 48)(8 36)(9 44)(10 32)(11 46)(12 34)(14 23)(16 19)(18 21)(20 41)(22 37)(24 39)(25 31)(26 45)(27 33)(28 47)(29 35)(30 43)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)(25 37)(26 42)(27 41)(28 40)(29 39)(30 38)
G:=sub<Sym(48)| (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,39)(8,40)(9,41)(10,42)(11,37)(12,38)(13,27)(14,28)(15,29)(16,30)(17,25)(18,26)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,31,23,44)(2,45,24,32)(3,33,19,46)(4,47,20,34)(5,35,21,48)(6,43,22,36)(7,18,29,42)(8,37,30,13)(9,14,25,38)(10,39,26,15)(11,16,27,40)(12,41,28,17), (1,38)(2,15)(3,40)(4,17)(5,42)(6,13)(7,48)(8,36)(9,44)(10,32)(11,46)(12,34)(14,23)(16,19)(18,21)(20,41)(22,37)(24,39)(25,31)(26,45)(27,33)(28,47)(29,35)(30,43), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48)(25,37)(26,42)(27,41)(28,40)(29,39)(30,38)>;
G:=Group( (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,39)(8,40)(9,41)(10,42)(11,37)(12,38)(13,27)(14,28)(15,29)(16,30)(17,25)(18,26)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,31,23,44)(2,45,24,32)(3,33,19,46)(4,47,20,34)(5,35,21,48)(6,43,22,36)(7,18,29,42)(8,37,30,13)(9,14,25,38)(10,39,26,15)(11,16,27,40)(12,41,28,17), (1,38)(2,15)(3,40)(4,17)(5,42)(6,13)(7,48)(8,36)(9,44)(10,32)(11,46)(12,34)(14,23)(16,19)(18,21)(20,41)(22,37)(24,39)(25,31)(26,45)(27,33)(28,47)(29,35)(30,43), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48)(25,37)(26,42)(27,41)(28,40)(29,39)(30,38) );
G=PermutationGroup([(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,39),(8,40),(9,41),(10,42),(11,37),(12,38),(13,27),(14,28),(15,29),(16,30),(17,25),(18,26),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,31,23,44),(2,45,24,32),(3,33,19,46),(4,47,20,34),(5,35,21,48),(6,43,22,36),(7,18,29,42),(8,37,30,13),(9,14,25,38),(10,39,26,15),(11,16,27,40),(12,41,28,17)], [(1,38),(2,15),(3,40),(4,17),(5,42),(6,13),(7,48),(8,36),(9,44),(10,32),(11,46),(12,34),(14,23),(16,19),(18,21),(20,41),(22,37),(24,39),(25,31),(26,45),(27,33),(28,47),(29,35),(30,43)], [(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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,15),(8,14),(9,13),(10,18),(11,17),(12,16),(19,47),(20,46),(21,45),(22,44),(23,43),(24,48),(25,37),(26,42),(27,41),(28,40),(29,39),(30,38)])
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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 2 | 0 |
| 0 | 0 | 0 | 12 | 0 | 2 |
| 0 | 0 | 12 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 2 | 4 | 11 | 9 |
| 0 | 0 | 9 | 11 | 4 | 2 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 12 | 12 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 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 |
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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,12,0,0,0,0,12,0,12,0,0,2,0,1,0,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,2,9,0,0,4,11,4,11,0,0,0,0,11,4,0,0,0,0,9,2],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,2,0,1,0,0,11,2,12,1],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 2N | ··· | 2U | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 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 | 6 | ··· | 6 | 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 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | 2+ (1+4) | D4⋊6D6 |
| kernel | C2×D4⋊6D6 | C2×C4○D12 | C2×S3×D4 | C2×D4⋊2S3 | D4⋊6D6 | C22×C3⋊D4 | D4×C2×C6 | C22×D4 | C22×C4 | C2×D4 | C24 | C6 | C2 |
| # reps | 1 | 2 | 4 | 4 | 16 | 4 | 1 | 1 | 1 | 12 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_6D_6
% in TeX
G:=Group("C2xD4:6D6"); // GroupNames label
G:=SmallGroup(192,1516);
// by ID
G=gap.SmallGroup(192,1516);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations