direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊D6, C12.33C24, D12.29C23, C3⋊C8⋊5C23, C4○D4⋊17D6, (C2×D4)⋊41D6, (C2×Q8)⋊33D6, C6⋊5(C8⋊C22), D4⋊5(C22×S3), (C3×D4)⋊5C23, Q8⋊6(C22×S3), (C3×Q8)⋊5C23, D4⋊S3⋊18C22, (C2×C12).217D4, C12.426(C2×D4), (C6×D4)⋊45C22, C4.33(S3×C23), (C6×Q8)⋊37C22, (C2×D12)⋊58C22, (C22×D12)⋊20C2, (C22×C4).297D6, C6.158(C22×D4), (C22×C6).122D4, (C2×C12).555C23, Q8⋊2S3⋊17C22, C23.75(C3⋊D4), C4.Dic3⋊36C22, (C22×C12).290C22, C3⋊6(C2×C8⋊C22), (C2×C4○D4)⋊6S3, (C6×C4○D4)⋊2C2, (C2×D4⋊S3)⋊31C2, (C2×C3⋊C8)⋊22C22, (C2×C6).75(C2×D4), C4.29(C2×C3⋊D4), (C2×Q8⋊2S3)⋊31C2, (C3×C4○D4)⋊16C22, (C2×C4).95(C3⋊D4), (C2×C4.Dic3)⋊30C2, C2.31(C22×C3⋊D4), (C2×C4).245(C22×S3), C22.118(C2×C3⋊D4), SmallGroup(192,1379)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 872 in 298 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], S3 [×4], C6, C6 [×2], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×2], D4 [×15], Q8 [×2], Q8, C23, C23 [×11], C12 [×2], C12 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×6], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C3⋊C8 [×4], D12 [×4], D12 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×S3 [×10], C22×C6, C22×C6, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], D4⋊S3 [×8], Q8⋊2S3 [×8], C2×D12 [×6], C2×D12 [×3], C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], S3×C23, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3 [×2], C2×Q8⋊2S3 [×2], D4⋊D6 [×8], C22×D12, C6×C4○D4, C2×D4⋊D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8⋊C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8⋊C22, D4⋊D6 [×2], C22×C3⋊D4, C2×D4⋊D6
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >
►On 48 points
(1 12)(2 10)(3 11)(4 21)(5 19)(6 20)(7 22)(8 23)(9 24)(13 18)(14 16)(15 17)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 18 5 8)(2 16 6 9)(3 17 4 7)(10 14 20 24)(11 15 21 22)(12 13 19 23)(25 46 28 43)(26 47 29 44)(27 48 30 45)(31 37 34 40)(32 38 35 41)(33 39 36 42)
(1 26)(2 30)(3 28)(4 25)(5 29)(6 27)(7 43)(8 47)(9 45)(10 39)(11 37)(12 41)(13 35)(14 33)(15 31)(16 48)(17 46)(18 44)(19 38)(20 42)(21 40)(22 34)(23 32)(24 36)
(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 11)(2 10)(3 12)(4 19)(5 21)(6 20)(7 13)(8 15)(9 14)(16 24)(17 23)(18 22)(25 35)(26 34)(27 33)(28 32)(29 31)(30 36)(37 47)(38 46)(39 45)(40 44)(41 43)(42 48)
G:=sub<Sym(48)| (1,12)(2,10)(3,11)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24)(13,18)(14,16)(15,17)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,18,5,8)(2,16,6,9)(3,17,4,7)(10,14,20,24)(11,15,21,22)(12,13,19,23)(25,46,28,43)(26,47,29,44)(27,48,30,45)(31,37,34,40)(32,38,35,41)(33,39,36,42), (1,26)(2,30)(3,28)(4,25)(5,29)(6,27)(7,43)(8,47)(9,45)(10,39)(11,37)(12,41)(13,35)(14,33)(15,31)(16,48)(17,46)(18,44)(19,38)(20,42)(21,40)(22,34)(23,32)(24,36), (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), (1,11)(2,10)(3,12)(4,19)(5,21)(6,20)(7,13)(8,15)(9,14)(16,24)(17,23)(18,22)(25,35)(26,34)(27,33)(28,32)(29,31)(30,36)(37,47)(38,46)(39,45)(40,44)(41,43)(42,48)>;
G:=Group( (1,12)(2,10)(3,11)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24)(13,18)(14,16)(15,17)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,18,5,8)(2,16,6,9)(3,17,4,7)(10,14,20,24)(11,15,21,22)(12,13,19,23)(25,46,28,43)(26,47,29,44)(27,48,30,45)(31,37,34,40)(32,38,35,41)(33,39,36,42), (1,26)(2,30)(3,28)(4,25)(5,29)(6,27)(7,43)(8,47)(9,45)(10,39)(11,37)(12,41)(13,35)(14,33)(15,31)(16,48)(17,46)(18,44)(19,38)(20,42)(21,40)(22,34)(23,32)(24,36), (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), (1,11)(2,10)(3,12)(4,19)(5,21)(6,20)(7,13)(8,15)(9,14)(16,24)(17,23)(18,22)(25,35)(26,34)(27,33)(28,32)(29,31)(30,36)(37,47)(38,46)(39,45)(40,44)(41,43)(42,48) );
G=PermutationGroup([(1,12),(2,10),(3,11),(4,21),(5,19),(6,20),(7,22),(8,23),(9,24),(13,18),(14,16),(15,17),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)], [(1,18,5,8),(2,16,6,9),(3,17,4,7),(10,14,20,24),(11,15,21,22),(12,13,19,23),(25,46,28,43),(26,47,29,44),(27,48,30,45),(31,37,34,40),(32,38,35,41),(33,39,36,42)], [(1,26),(2,30),(3,28),(4,25),(5,29),(6,27),(7,43),(8,47),(9,45),(10,39),(11,37),(12,41),(13,35),(14,33),(15,31),(16,48),(17,46),(18,44),(19,38),(20,42),(21,40),(22,34),(23,32),(24,36)], [(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)], [(1,11),(2,10),(3,12),(4,19),(5,21),(6,20),(7,13),(8,15),(9,14),(16,24),(17,23),(18,22),(25,35),(26,34),(27,33),(28,32),(29,31),(30,36),(37,47),(38,46),(39,45),(40,44),(41,43),(42,48)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 14 | 0 | 0 |
| 0 | 0 | 59 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 59 |
| 0 | 0 | 0 | 0 | 14 | 66 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 13 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 14 |
| 0 | 0 | 0 | 0 | 59 | 7 |
| 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 14 |
| 0 | 0 | 0 | 0 | 7 | 7 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,0,0,0,0,7,14,0,0,0,0,59,66],[43,13,0,0,0,0,60,30,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,66,59,0,0,0,0,14,7,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,66,7,0,0,0,0,14,7] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | ··· | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8⋊C22 | D4⋊D6 |
| kernel | C2×D4⋊D6 | C2×C4.Dic3 | C2×D4⋊S3 | C2×Q8⋊2S3 | D4⋊D6 | C22×D12 | C6×C4○D4 | C2×C4○D4 | C2×C12 | C22×C6 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 4 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_4\rtimes D_6
% in TeX
G:=Group("C2xD4:D6"); // GroupNames label
G:=SmallGroup(192,1379);
// by ID
G=gap.SmallGroup(192,1379);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,1684,235,102,6278]);
// 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=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations