direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6⋊D4, C23⋊6D12, C24.65D6, C6⋊1C22≀C2, D6⋊10(C2×D4), (C22×C6)⋊9D4, (S3×C24)⋊1C2, (C2×C12)⋊3C23, C22⋊C4⋊37D6, C22⋊4(C2×D12), (C22×C4)⋊10D6, C6.6(C22×D4), D6⋊C4⋊44C22, (C22×D12)⋊5C2, (C22×S3)⋊13D4, (C2×C6).31C24, C2.8(C22×D12), (C2×D12)⋊43C22, (C22×S3)⋊1C23, (S3×C23)⋊3C22, (C22×C12)⋊7C22, (C2×Dic3)⋊1C23, C22.125(S3×D4), C22.70(S3×C23), (C23×C6).57C22, (C22×C6).123C23, C23.156(C22×S3), (C22×Dic3)⋊6C22, C2.8(C2×S3×D4), (C2×C6)⋊4(C2×D4), C3⋊1(C2×C22≀C2), (C2×D6⋊C4)⋊16C2, (C2×C4)⋊3(C22×S3), (C2×C22⋊C4)⋊10S3, (C6×C22⋊C4)⋊13C2, (C22×C3⋊D4)⋊3C2, (C2×C3⋊D4)⋊34C22, (C3×C22⋊C4)⋊46C22, SmallGroup(192,1046)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D6⋊D4
G = < a,b,c,d,e | a2=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b3c, ece=bc, ede=d-1 >
Subgroups: 2088 in 662 conjugacy classes, 143 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C22×D4, C25, D6⋊C4, C3×C22⋊C4, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, S3×C23, S3×C23, C23×C6, C2×C22≀C2, D6⋊D4, C2×D6⋊C4, C6×C22⋊C4, C22×D12, C22×C3⋊D4, S3×C24, C2×D6⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22≀C2, C22×D4, C2×D12, S3×D4, S3×C23, C2×C22≀C2, D6⋊D4, C22×D12, C2×S3×D4, C2×D6⋊D4
►On 48 points
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 18)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)(31 40)(32 39)(33 38)(34 37)(35 42)(36 41)
(1 44 18 8)(2 45 13 9)(3 46 14 10)(4 47 15 11)(5 48 16 12)(6 43 17 7)(19 39 27 31)(20 40 28 32)(21 41 29 33)(22 42 30 34)(23 37 25 35)(24 38 26 36)
(1 25)(2 30)(3 29)(4 28)(5 27)(6 26)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)(37 44)(38 43)(39 48)(40 47)(41 46)(42 45)
G:=sub<Sym(48)| (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(19,25)(20,30)(21,29)(22,28)(23,27)(24,26)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41), (1,44,18,8)(2,45,13,9)(3,46,14,10)(4,47,15,11)(5,48,16,12)(6,43,17,7)(19,39,27,31)(20,40,28,32)(21,41,29,33)(22,42,30,34)(23,37,25,35)(24,38,26,36), (1,25)(2,30)(3,29)(4,28)(5,27)(6,26)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)(37,44)(38,43)(39,48)(40,47)(41,46)(42,45)>;
G:=Group( (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(19,25)(20,30)(21,29)(22,28)(23,27)(24,26)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41), (1,44,18,8)(2,45,13,9)(3,46,14,10)(4,47,15,11)(5,48,16,12)(6,43,17,7)(19,39,27,31)(20,40,28,32)(21,41,29,33)(22,42,30,34)(23,37,25,35)(24,38,26,36), (1,25)(2,30)(3,29)(4,28)(5,27)(6,26)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)(37,44)(38,43)(39,48)(40,47)(41,46)(42,45) );
G=PermutationGroup([[(1,40),(2,41),(3,42),(4,37),(5,38),(6,39),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,33),(14,34),(15,35),(16,36),(17,31),(18,32),(25,47),(26,48),(27,43),(28,44),(29,45),(30,46)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,18),(7,47),(8,46),(9,45),(10,44),(11,43),(12,48),(19,25),(20,30),(21,29),(22,28),(23,27),(24,26),(31,40),(32,39),(33,38),(34,37),(35,42),(36,41)], [(1,44,18,8),(2,45,13,9),(3,46,14,10),(4,47,15,11),(5,48,16,12),(6,43,17,7),(19,39,27,31),(20,40,28,32),(21,41,29,33),(22,42,30,34),(23,37,25,35),(24,38,26,36)], [(1,25),(2,30),(3,29),(4,28),(5,27),(6,26),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,22),(14,21),(15,20),(16,19),(17,24),(18,23),(37,44),(38,43),(39,48),(40,47),(41,46),(42,45)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 2T | 2U | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | S3×D4 |
| kernel | C2×D6⋊D4 | D6⋊D4 | C2×D6⋊C4 | C6×C22⋊C4 | C22×D12 | C22×C3⋊D4 | S3×C24 | C2×C22⋊C4 | C22×S3 | C22×C6 | C22⋊C4 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 8 | 2 | 1 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 2 | 1 | 8 | 4 |
Matrix representation of C2×D6⋊D4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,3,7,0,0,0,6,10,0,0,0,0,0,0,1,0,0,0,12,0],[1,0,0,0,0,0,3,3,0,0,0,6,10,0,0,0,0,0,0,1,0,0,0,1,0] >;
C2×D6⋊D4 in GAP, Magma, Sage, TeX
C_2\times D_6\rtimes D_4
% in TeX
G:=Group("C2xD6:D4"); // GroupNames label
G:=SmallGroup(192,1046);
// by ID
G=gap.SmallGroup(192,1046);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^4=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^3*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations