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, C8⋊D6⋊4C2, Q8⋊3D6⋊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, C24⋊C2⋊7C22, C8⋊S3⋊7C22, D4⋊S3⋊16C22, Q8.13D6⋊5C2, C12.246(C2×D4), C4○D12⋊9C22, C4.25(S3×C23), (C6×Q8)⋊22C22, (C3×Q16)⋊3C22, (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, (C3×D4).18C23, D4.18(C22×S3), C3⋊Q16⋊14C22, 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
Generators and relations for C24.C23
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 >
Subgroups: 736 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, D12, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C22×S3, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, S3×C8, C8⋊S3, C24⋊C2, D24, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, S3×D4, S3×Q8, S3×Q8, Q8⋊3S3, Q8⋊3S3, C6×Q8, C3×C4○D4, D4○SD16, D12.C4, C8⋊D6, S3×SD16, Q8⋊3D6, Q16⋊S3, D24⋊C2, C2×Q8⋊2S3, Q8.13D6, C3×C8.C22, Q8.15D6, D4○D12, C24.C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D4○SD16, C2×S3×D4, C24.C23
►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 39 13 27)(2 46 14 34)(3 29 15 41)(4 36 16 48)(5 43 17 31)(6 26 18 38)(7 33 19 45)(8 40 20 28)(9 47 21 35)(10 30 22 42)(11 37 23 25)(12 44 24 32)
(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 34 37 46)(26 29 38 41)(27 48 39 36)(28 43 40 31)(30 33 42 45)(32 47 44 35)
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), (1,39,13,27)(2,46,14,34)(3,29,15,41)(4,36,16,48)(5,43,17,31)(6,26,18,38)(7,33,19,45)(8,40,20,28)(9,47,21,35)(10,30,22,42)(11,37,23,25)(12,44,24,32), (1,4,13,16)(2,23,14,11)(3,18,15,6)(5,8,17,20)(7,22,19,10)(9,12,21,24)(25,34,37,46)(26,29,38,41)(27,48,39,36)(28,43,40,31)(30,33,42,45)(32,47,44,35)>;
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), (1,39,13,27)(2,46,14,34)(3,29,15,41)(4,36,16,48)(5,43,17,31)(6,26,18,38)(7,33,19,45)(8,40,20,28)(9,47,21,35)(10,30,22,42)(11,37,23,25)(12,44,24,32), (1,4,13,16)(2,23,14,11)(3,18,15,6)(5,8,17,20)(7,22,19,10)(9,12,21,24)(25,34,37,46)(26,29,38,41)(27,48,39,36)(28,43,40,31)(30,33,42,45)(32,47,44,35) );
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)], [(1,39,13,27),(2,46,14,34),(3,29,15,41),(4,36,16,48),(5,43,17,31),(6,26,18,38),(7,33,19,45),(8,40,20,28),(9,47,21,35),(10,30,22,42),(11,37,23,25),(12,44,24,32)], [(1,4,13,16),(2,23,14,11),(3,18,15,6),(5,8,17,20),(7,22,19,10),(9,12,21,24),(25,34,37,46),(26,29,38,41),(27,48,39,36),(28,43,40,31),(30,33,42,45),(32,47,44,35)]])
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 |
Matrix representation of C24.C23 ►in 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] >;
C24.C23 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