metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1C4≀C2, (C2×D12)⋊1C4, C4⋊Dic3⋊1C4, (C2×C6).29D8, (C2×C12).221D4, (C2×C4).103D12, C12⋊7D4.7C2, C6.1(C23⋊C4), (C2×C6).37SD16, (C22×C4).68D6, C6.1(D4⋊C4), C12.55D4⋊1C2, C2.C42⋊7S3, C22.7(D4⋊S3), C2.3(C6.D8), (C22×C6).174D4, C3⋊1(C22.SD16), C2.4(C42⋊4S3), C22.54(D6⋊C4), C23.77(C3⋊D4), (C22×C12).89C22, C22.7(Q8⋊2S3), C2.4(C23.6D6), (C2×C4).8(C4×S3), (C2×C12).20(C2×C4), (C2×C6).33(C22⋊C4), (C3×C2.C42)⋊13C2, SmallGroup(192,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.C4≀C2
G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, cac=a-1, ad=da, cbc=b-1, dbd-1=a3b, dcd-1=a3b-1c >
Subgroups: 320 in 90 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2.C42, C22⋊C8, C4⋊D4, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C22.SD16, C12.55D4, C3×C2.C42, C12⋊7D4, C6.C4≀C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, C23⋊C4, D4⋊C4, C4≀C2, D6⋊C4, D4⋊S3, Q8⋊2S3, C22.SD16, C42⋊4S3, C23.6D6, C6.D8, C6.C4≀C2
►On 48 points
(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 19 10 16)(2 20 11 17)(3 21 12 18)(4 22 7 13)(5 23 8 14)(6 24 9 15)(25 40 34 43)(26 41 35 44)(27 42 36 45)(28 37 31 46)(29 38 32 47)(30 39 33 48)
(1 44)(2 43)(3 48)(4 47)(5 46)(6 45)(7 38)(8 37)(9 42)(10 41)(11 40)(12 39)(13 29)(14 28)(15 27)(16 26)(17 25)(18 30)(19 35)(20 34)(21 33)(22 32)(23 31)(24 36)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
G:=sub<Sym(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), (1,19,10,16)(2,20,11,17)(3,21,12,18)(4,22,7,13)(5,23,8,14)(6,24,9,15)(25,40,34,43)(26,41,35,44)(27,42,36,45)(28,37,31,46)(29,38,32,47)(30,39,33,48), (1,44)(2,43)(3,48)(4,47)(5,46)(6,45)(7,38)(8,37)(9,42)(10,41)(11,40)(12,39)(13,29)(14,28)(15,27)(16,26)(17,25)(18,30)(19,35)(20,34)(21,33)(22,32)(23,31)(24,36), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)>;
G:=Group( (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), (1,19,10,16)(2,20,11,17)(3,21,12,18)(4,22,7,13)(5,23,8,14)(6,24,9,15)(25,40,34,43)(26,41,35,44)(27,42,36,45)(28,37,31,46)(29,38,32,47)(30,39,33,48), (1,44)(2,43)(3,48)(4,47)(5,46)(6,45)(7,38)(8,37)(9,42)(10,41)(11,40)(12,39)(13,29)(14,28)(15,27)(16,26)(17,25)(18,30)(19,35)(20,34)(21,33)(22,32)(23,31)(24,36), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39) );
G=PermutationGroup([[(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)], [(1,19,10,16),(2,20,11,17),(3,21,12,18),(4,22,7,13),(5,23,8,14),(6,24,9,15),(25,40,34,43),(26,41,35,44),(27,42,36,45),(28,37,31,46),(29,38,32,47),(30,39,33,48)], [(1,44),(2,43),(3,48),(4,47),(5,46),(6,45),(7,38),(8,37),(9,42),(10,41),(11,40),(12,39),(13,29),(14,28),(15,27),(16,26),(17,25),(18,30),(19,35),(20,34),(21,33),(22,32),(23,31),(24,36)], [(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | C42⋊4S3 | C23⋊C4 | D4⋊S3 | Q8⋊2S3 | C23.6D6 |
| kernel | C6.C4≀C2 | C12.55D4 | C3×C2.C42 | C12⋊7D4 | C4⋊Dic3 | C2×D12 | C2.C42 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C6 | C2 | C6 | C22 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of C6.C4≀C2 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 10 | 64 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 67 | 27 |
| 16 | 16 | 0 | 0 |
| 16 | 57 | 0 | 0 |
| 0 | 0 | 67 | 54 |
| 0 | 0 | 71 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 68 | 46 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,8,10,0,0,0,64],[0,1,0,0,72,0,0,0,0,0,46,67,0,0,0,27],[16,16,0,0,16,57,0,0,0,0,67,71,0,0,54,6],[1,0,0,0,0,72,0,0,0,0,1,68,0,0,0,46] >;
C6.C4≀C2 in GAP, Magma, Sage, TeX
C_6.C_4\wr C_2
% in TeX
G:=Group("C6.C4wrC2"); // GroupNames label
G:=SmallGroup(192,10);
// by ID
G=gap.SmallGroup(192,10);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,1571,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^2=d^4=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a^3*b,d*c*d^-1=a^3*b^-1*c>;
// generators/relations