metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊5D6, SD16⋊7D6, D12.43D4, D24⋊4C22, C24.6C23, M4(2)⋊13D6, C12.25C24, Dic6.43D4, D12.18C23, Dic6.18C23, D4○D12⋊8C2, Q8⋊3D6⋊4C2, C8⋊D6⋊4C2, (C2×Q8)⋊15D6, C3⋊D4.6D4, D12.C4⋊3C2, (S3×C8)⋊6C22, C4○D4.30D6, (S3×SD16)⋊4C2, D6.34(C2×D4), C3⋊5(D4○SD16), Q16⋊S3⋊3C2, C4.117(S3×D4), C8.C22⋊4S3, C3⋊C8.27C23, C8.6(C22×S3), D24⋊C2⋊2C2, (S3×Q8)⋊4C22, C8⋊S3⋊7C22, C24⋊C2⋊7C22, D4⋊S3⋊16C22, Q8.13D6⋊5C2, C12.246(C2×D4), C4○D12⋊9C22, C4.25(S3×C23), (C3×Q16)⋊3C22, (C6×Q8)⋊22C22, (S3×D4).4C22, C22.16(S3×D4), (C4×S3).16C23, D4.S3⋊15C22, Q8.15D6⋊5C2, Dic3.39(C2×D4), Q8⋊3S3⋊4C22, (C3×SD16)⋊7C22, D4.18(C22×S3), C3⋊Q16⋊14C22, (C3×D4).18C23, C6.126(C22×D4), Q8.28(C22×S3), (C3×Q8).18C23, (C2×C12).116C23, Q8⋊2S3⋊15C22, (C3×M4(2))⋊7C22, (C2×D12).182C22, C2.99(C2×S3×D4), (C2×C3⋊C8)⋊19C22, (C2×C6).71(C2×D4), (C3×C8.C22)⋊3C2, (C2×Q8⋊2S3)⋊29C2, (C2×C4).100(C22×S3), (C3×C4○D4).27C22, SmallGroup(192,1337)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 736 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×6], C22, C22 [×9], S3 [×5], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×15], Q8, Q8 [×2], Q8 [×5], C23 [×3], Dic3 [×2], Dic3, C12 [×2], C12 [×3], D6 [×2], D6 [×6], C2×C6, C2×C6, C2×C8 [×3], M4(2), M4(2) [×2], D8 [×3], SD16 [×2], SD16 [×8], Q16 [×2], Q16, C2×D4 [×6], C2×Q8, C2×Q8 [×3], C4○D4, C4○D4 [×10], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×2], C4×S3 [×2], C4×S3 [×7], D12 [×2], D12 [×2], D12 [×5], C3⋊D4 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C3×Q8, C22×S3 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22, C8.C22 [×2], 2+ (1+4), 2- (1+4), S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24 [×2], C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, Q8⋊2S3 [×4], C3⋊Q16, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C2×D12, C2×D12, C4○D12 [×2], C4○D12 [×3], S3×D4 [×2], S3×D4 [×2], S3×Q8 [×2], S3×Q8, Q8⋊3S3 [×4], Q8⋊3S3, C6×Q8, C3×C4○D4, D4○SD16, D12.C4, C8⋊D6, S3×SD16 [×2], Q8⋊3D6 [×2], Q16⋊S3 [×2], D24⋊C2 [×2], C2×Q8⋊2S3, Q8.13D6, C3×C8.C22, Q8.15D6, D4○D12, C24.C23
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], S3×C23, D4○SD16, C2×S3×D4, C24.C23
Generators and relations
G = < a,b,c,d | a24=b2=1, c2=d2=a12, bab=a5, cac-1=a7, dad-1=a19, bc=cb, dbd-1=a12b, cd=dc >
►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)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 41)(26 46)(28 32)(29 37)(30 42)(31 47)(34 38)(35 43)(36 48)(40 44)
(1 45 13 33)(2 28 14 40)(3 35 15 47)(4 42 16 30)(5 25 17 37)(6 32 18 44)(7 39 19 27)(8 46 20 34)(9 29 21 41)(10 36 22 48)(11 43 23 31)(12 26 24 38)
(1 4 13 16)(2 23 14 11)(3 18 15 6)(5 8 17 20)(7 22 19 10)(9 12 21 24)(25 46 37 34)(26 41 38 29)(27 36 39 48)(28 31 40 43)(30 45 42 33)(32 35 44 47)
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), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,41)(26,46)(28,32)(29,37)(30,42)(31,47)(34,38)(35,43)(36,48)(40,44), (1,45,13,33)(2,28,14,40)(3,35,15,47)(4,42,16,30)(5,25,17,37)(6,32,18,44)(7,39,19,27)(8,46,20,34)(9,29,21,41)(10,36,22,48)(11,43,23,31)(12,26,24,38), (1,4,13,16)(2,23,14,11)(3,18,15,6)(5,8,17,20)(7,22,19,10)(9,12,21,24)(25,46,37,34)(26,41,38,29)(27,36,39,48)(28,31,40,43)(30,45,42,33)(32,35,44,47)>;
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), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,41)(26,46)(28,32)(29,37)(30,42)(31,47)(34,38)(35,43)(36,48)(40,44), (1,45,13,33)(2,28,14,40)(3,35,15,47)(4,42,16,30)(5,25,17,37)(6,32,18,44)(7,39,19,27)(8,46,20,34)(9,29,21,41)(10,36,22,48)(11,43,23,31)(12,26,24,38), (1,4,13,16)(2,23,14,11)(3,18,15,6)(5,8,17,20)(7,22,19,10)(9,12,21,24)(25,46,37,34)(26,41,38,29)(27,36,39,48)(28,31,40,43)(30,45,42,33)(32,35,44,47) );
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)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,41),(26,46),(28,32),(29,37),(30,42),(31,47),(34,38),(35,43),(36,48),(40,44)], [(1,45,13,33),(2,28,14,40),(3,35,15,47),(4,42,16,30),(5,25,17,37),(6,32,18,44),(7,39,19,27),(8,46,20,34),(9,29,21,41),(10,36,22,48),(11,43,23,31),(12,26,24,38)], [(1,4,13,16),(2,23,14,11),(3,18,15,6),(5,8,17,20),(7,22,19,10),(9,12,21,24),(25,46,37,34),(26,41,38,29),(27,36,39,48),(28,31,40,43),(30,45,42,33),(32,35,44,47)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 61 |
| 0 | 0 | 6 | 0 | 0 | 61 |
| 0 | 0 | 0 | 67 | 6 | 67 |
| 0 | 0 | 67 | 6 | 6 | 67 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 72 | 0 |
| 0 | 0 | 1 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 61 | 0 | 0 |
| 0 | 0 | 67 | 0 | 0 | 0 |
| 0 | 0 | 0 | 67 | 67 | 6 |
| 0 | 0 | 67 | 67 | 6 | 6 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 61 |
| 0 | 0 | 67 | 0 | 12 | 0 |
| 0 | 0 | 0 | 6 | 6 | 67 |
| 0 | 0 | 67 | 6 | 6 | 67 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,6,0,67,0,0,0,0,67,6,0,0,12,0,6,6,0,0,61,61,67,67],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,67,0,67,0,0,61,0,67,67,0,0,0,0,67,6,0,0,0,0,6,6],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,67,0,67,0,0,0,0,6,6,0,0,12,12,6,6,0,0,61,0,67,67] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 2 | 4 | 8 | 4 | 4 | 6 | 6 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | S3×D4 | S3×D4 | D4○SD16 | C24.C23 |
| kernel | C24.C23 | D12.C4 | C8⋊D6 | S3×SD16 | Q8⋊3D6 | Q16⋊S3 | D24⋊C2 | C2×Q8⋊2S3 | Q8.13D6 | C3×C8.C22 | Q8.15D6 | D4○D12 | C8.C22 | Dic6 | D12 | C3⋊D4 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 |
In GAP, Magma, Sage, TeX
C_{24}.C_2^3 % in TeX
G:=Group("C24.C2^3"); // GroupNames label
G:=SmallGroup(192,1337);
// by ID
G=gap.SmallGroup(192,1337);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,184,570,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^24=b^2=1,c^2=d^2=a^12,b*a*b=a^5,c*a*c^-1=a^7,d*a*d^-1=a^19,b*c=c*b,d*b*d^-1=a^12*b,c*d=d*c>;
// generators/relations