metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D24, C23.36D12, C6.9C4≀C2, (C2×C6).1D8, (C2×D12)⋊2C4, C22⋊C8⋊2S3, C4⋊Dic3⋊4C4, (C2×C6).2SD16, C12⋊7D4.1C2, (C2×C12).440D4, C6.6(C23⋊C4), (C22×C4).71D6, (C22×C6).40D4, C2.3(C2.D24), C3⋊2(C22.SD16), C2.7(D12⋊C4), C6.11(D4⋊C4), C6.C42⋊26C2, C22.2(C24⋊C2), C22.59(D6⋊C4), (C22×C12).41C22, C2.7(C23.6D6), (C3×C22⋊C8)⋊2C2, (C2×C4).13(C4×S3), (C2×C12).25(C2×C4), (C2×C4).211(C3⋊D4), (C2×C6).42(C22⋊C4), SmallGroup(192,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.2D24
G = < a,b,c,d | a2=b2=c24=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Subgroups: 352 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, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2.C42, C22⋊C8, C4⋊D4, C4⋊Dic3, D6⋊C4, C2×C24, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C22.SD16, C6.C42, C3×C22⋊C8, C12⋊7D4, C22.2D24
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, C24⋊C2, D24, D6⋊C4, C22.SD16, C23.6D6, C2.D24, D12⋊C4, C22.2D24
►On 48 points
(1 29)(3 31)(5 33)(7 35)(9 37)(11 39)(13 41)(15 43)(17 45)(19 47)(21 25)(23 27)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 25)(22 26)(23 27)(24 28)
(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 15 29 43)(2 14)(3 41 31 13)(4 40)(5 11 33 39)(6 10)(7 37 35 9)(8 36)(12 32)(16 28)(17 23 45 27)(18 22)(19 25 47 21)(20 48)(24 44)(26 46)(30 42)(34 38)
G:=sub<Sym(48)| (1,29)(3,31)(5,33)(7,35)(9,37)(11,39)(13,41)(15,43)(17,45)(19,47)(21,25)(23,27), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,25)(22,26)(23,27)(24,28), (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), (1,15,29,43)(2,14)(3,41,31,13)(4,40)(5,11,33,39)(6,10)(7,37,35,9)(8,36)(12,32)(16,28)(17,23,45,27)(18,22)(19,25,47,21)(20,48)(24,44)(26,46)(30,42)(34,38)>;
G:=Group( (1,29)(3,31)(5,33)(7,35)(9,37)(11,39)(13,41)(15,43)(17,45)(19,47)(21,25)(23,27), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,25)(22,26)(23,27)(24,28), (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), (1,15,29,43)(2,14)(3,41,31,13)(4,40)(5,11,33,39)(6,10)(7,37,35,9)(8,36)(12,32)(16,28)(17,23,45,27)(18,22)(19,25,47,21)(20,48)(24,44)(26,46)(30,42)(34,38) );
G=PermutationGroup([[(1,29),(3,31),(5,33),(7,35),(9,37),(11,39),(13,41),(15,43),(17,45),(19,47),(21,25),(23,27)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,25),(22,26),(23,27),(24,28)], [(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)], [(1,15,29,43),(2,14),(3,41,31,13),(4,40),(5,11,33,39),(6,10),(7,37,35,9),(8,36),(12,32),(16,28),(17,23,45,27),(18,22),(19,25,47,21),(20,48),(24,44),(26,46),(30,42),(34,38)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | C3⋊D4 | D12 | C4≀C2 | C24⋊C2 | D24 | C23⋊C4 | C23.6D6 | D12⋊C4 |
| kernel | C22.2D24 | C6.C42 | C3×C22⋊C8 | C12⋊7D4 | C4⋊Dic3 | C2×D12 | C22⋊C8 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C6 | C22 | C22 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of C22.2D24 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 55 | 23 |
| 0 | 0 | 50 | 5 |
| 27 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 66 | 14 |
| 0 | 0 | 7 | 7 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[0,46,0,0,1,0,0,0,0,0,55,50,0,0,23,5],[27,0,0,0,0,72,0,0,0,0,66,7,0,0,14,7] >;
C22.2D24 in GAP, Magma, Sage, TeX
C_2^2._2D_{24} % in TeX
G:=Group("C2^2.2D24"); // GroupNames label
G:=SmallGroup(192,29);
// by ID
G=gap.SmallGroup(192,29);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,387,100,1123,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^24=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations