metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C24)⋊1C4, (C2×C8)⋊1Dic3, (C2×C4).14D12, C4.Dic3⋊6C4, C12.11(C4⋊C4), (C2×C12).11Q8, (C2×C4).5Dic6, C4.35(D6⋊C4), (C2×C12).109D4, (C2×C6).20C42, C23.14(C4×S3), (C22×C4).78D6, C4.13(C4⋊Dic3), C6.D4.1C4, (C2×M4(2)).7S3, C12.11(C22⋊C4), C3⋊2(M4(2)⋊4C4), C4.11(Dic3⋊C4), C22.19(D6⋊C4), (C6×M4(2)).11C2, C22.10(C4×Dic3), C4.28(C6.D4), C22.6(Dic3⋊C4), C23.26D6.9C2, C6.15(C2.C42), C2.15(C6.C42), (C22×C12).125C22, (C2×C6).8(C4⋊C4), (C2×C4).21(C4×S3), (C2×C12).304(C2×C4), (C22×C6).32(C2×C4), (C2×C4).76(C2×Dic3), (C2×C4).180(C3⋊D4), (C2×C6).10(C22⋊C4), (C2×C4.Dic3).11C2, SmallGroup(192,115)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C24)⋊C4
G = < a,b,c | a2=b24=c4=1, ab=ba, cac-1=ab12, cbc-1=ab5 >
Subgroups: 200 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×C12, M4(2)⋊4C4, C2×C4.Dic3, C23.26D6, C6×M4(2), (C2×C24)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, M4(2)⋊4C4, C6.C42, (C2×C24)⋊C4
►On 48 points
(1 47)(2 48)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 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)
(2 40 14 28)(3 11)(4 26 16 38)(5 21)(6 36 18 48)(8 46 20 34)(9 17)(10 32 22 44)(12 42 24 30)(15 23)(25 45)(27 31)(29 41)(33 37)(35 47)(39 43)
G:=sub<Sym(48)| (1,47)(2,48)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (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), (2,40,14,28)(3,11)(4,26,16,38)(5,21)(6,36,18,48)(8,46,20,34)(9,17)(10,32,22,44)(12,42,24,30)(15,23)(25,45)(27,31)(29,41)(33,37)(35,47)(39,43)>;
G:=Group( (1,47)(2,48)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (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), (2,40,14,28)(3,11)(4,26,16,38)(5,21)(6,36,18,48)(8,46,20,34)(9,17)(10,32,22,44)(12,42,24,30)(15,23)(25,45)(27,31)(29,41)(33,37)(35,47)(39,43) );
G=PermutationGroup([[(1,47),(2,48),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46)], [(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)], [(2,40,14,28),(3,11),(4,26,16,38),(5,21),(6,36,18,48),(8,46,20,34),(9,17),(10,32,22,44),(12,42,24,30),(15,23),(25,45),(27,31),(29,41),(33,37),(35,47),(39,43)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | C4×S3 | M4(2)⋊4C4 | (C2×C24)⋊C4 |
| kernel | (C2×C24)⋊C4 | C2×C4.Dic3 | C23.26D6 | C6×M4(2) | C4.Dic3 | C6.D4 | C2×C24 | C2×M4(2) | C2×C12 | C2×C12 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of (C2×C24)⋊C4 ►in GL4(𝔽73) generated by
| 43 | 60 | 0 | 0 |
| 13 | 30 | 0 | 0 |
| 0 | 0 | 30 | 60 |
| 0 | 0 | 13 | 43 |
| 36 | 69 | 46 | 44 |
| 4 | 32 | 2 | 46 |
| 36 | 55 | 41 | 4 |
| 55 | 36 | 69 | 37 |
| 1 | 72 | 0 | 2 |
| 0 | 72 | 2 | 2 |
| 0 | 0 | 60 | 30 |
| 0 | 0 | 43 | 13 |
G:=sub<GL(4,GF(73))| [43,13,0,0,60,30,0,0,0,0,30,13,0,0,60,43],[36,4,36,55,69,32,55,36,46,2,41,69,44,46,4,37],[1,0,0,0,72,72,0,0,0,2,60,43,2,2,30,13] >;
(C2×C24)⋊C4 in GAP, Magma, Sage, TeX
(C_2\times C_{24})\rtimes C_4 % in TeX
G:=Group("(C2xC24):C4"); // GroupNames label
G:=SmallGroup(192,115);
// by ID
G=gap.SmallGroup(192,115);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^24=c^4=1,a*b=b*a,c*a*c^-1=a*b^12,c*b*c^-1=a*b^5>;
// generators/relations