metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.9C23, D24⋊10C22, M4(2)⋊19D6, C12.60C24, C23.26D12, Dic12⋊9C22, D12.23C23, Dic6.23C23, (C2×C8)⋊5D6, C4○D24⋊10C2, C8⋊D6⋊13C2, (C2×C24)⋊8C22, C4.73(C2×D12), C8.9(C22×S3), (C2×C4).157D12, (C2×C12).205D4, C8.D6⋊13C2, C12.239(C2×D4), (C2×M4(2))⋊5S3, (C6×M4(2))⋊5C2, C4.57(S3×C23), C6.27(C22×D4), C24⋊C2⋊10C22, C4○D12⋊17C22, (C2×D12)⋊53C22, C3⋊1(D8⋊C22), C22.22(C2×D12), C2.29(C22×D12), (C22×C6).120D4, (C22×C4).283D6, (C2×C12).798C23, (C2×Dic6)⋊64C22, (C3×M4(2))⋊21C22, (C22×C12).268C22, (C2×C6).64(C2×D4), (C2×C4○D12)⋊27C2, (C2×C4).225(C22×S3), SmallGroup(192,1307)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.9C23
G = < a,b,c,d | a24=b2=1, c2=d2=a12, bab=a11, ac=ca, dad-1=a13, bc=cb, bd=db, cd=dc >
Subgroups: 728 in 262 conjugacy classes, 107 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C24⋊C2, D24, Dic12, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, D8⋊C22, C4○D24, C8⋊D6, C8.D6, C6×M4(2), C2×C4○D12, C24.9C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C2×D12, S3×C23, D8⋊C22, C22×D12, C24.9C23
►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 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 37)(26 48)(27 35)(28 46)(29 33)(30 44)(32 42)(34 40)(36 38)(39 47)(41 45)
(1 31 13 43)(2 32 14 44)(3 33 15 45)(4 34 16 46)(5 35 17 47)(6 36 18 48)(7 37 19 25)(8 38 20 26)(9 39 21 27)(10 40 22 28)(11 41 23 29)(12 42 24 30)
(1 43 13 31)(2 32 14 44)(3 45 15 33)(4 34 16 46)(5 47 17 35)(6 36 18 48)(7 25 19 37)(8 38 20 26)(9 27 21 39)(10 40 22 28)(11 29 23 41)(12 42 24 30)
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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,37)(26,48)(27,35)(28,46)(29,33)(30,44)(32,42)(34,40)(36,38)(39,47)(41,45), (1,31,13,43)(2,32,14,44)(3,33,15,45)(4,34,16,46)(5,35,17,47)(6,36,18,48)(7,37,19,25)(8,38,20,26)(9,39,21,27)(10,40,22,28)(11,41,23,29)(12,42,24,30), (1,43,13,31)(2,32,14,44)(3,45,15,33)(4,34,16,46)(5,47,17,35)(6,36,18,48)(7,25,19,37)(8,38,20,26)(9,27,21,39)(10,40,22,28)(11,29,23,41)(12,42,24,30)>;
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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,37)(26,48)(27,35)(28,46)(29,33)(30,44)(32,42)(34,40)(36,38)(39,47)(41,45), (1,31,13,43)(2,32,14,44)(3,33,15,45)(4,34,16,46)(5,35,17,47)(6,36,18,48)(7,37,19,25)(8,38,20,26)(9,39,21,27)(10,40,22,28)(11,41,23,29)(12,42,24,30), (1,43,13,31)(2,32,14,44)(3,45,15,33)(4,34,16,46)(5,47,17,35)(6,36,18,48)(7,25,19,37)(8,38,20,26)(9,27,21,39)(10,40,22,28)(11,29,23,41)(12,42,24,30) );
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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,37),(26,48),(27,35),(28,46),(29,33),(30,44),(32,42),(34,40),(36,38),(39,47),(41,45)], [(1,31,13,43),(2,32,14,44),(3,33,15,45),(4,34,16,46),(5,35,17,47),(6,36,18,48),(7,37,19,25),(8,38,20,26),(9,39,21,27),(10,40,22,28),(11,41,23,29),(12,42,24,30)], [(1,43,13,31),(2,32,14,44),(3,45,15,33),(4,34,16,46),(5,47,17,35),(6,36,18,48),(7,25,19,37),(8,38,20,26),(9,27,21,39),(10,40,22,28),(11,29,23,41),(12,42,24,30)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 2 | 2 | 3 | 4 | 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 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | D12 | D8⋊C22 | C24.9C23 |
| kernel | C24.9C23 | C4○D24 | C8⋊D6 | C8.D6 | C6×M4(2) | C2×C4○D12 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 3 | 1 | 2 | 4 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C24.9C23 ►in GL4(𝔽73) generated by
| 0 | 0 | 46 | 46 |
| 0 | 0 | 27 | 0 |
| 43 | 30 | 0 | 0 |
| 43 | 13 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 59 | 66 |
| 0 | 0 | 7 | 14 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [0,0,43,43,0,0,30,13,46,27,0,0,46,0,0,0],[0,72,0,0,72,0,0,0,0,0,59,7,0,0,66,14],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[46,0,0,0,0,46,0,0,0,0,27,0,0,0,0,27] >;
C24.9C23 in GAP, Magma, Sage, TeX
C_{24}._9C_2^3 % in TeX
G:=Group("C24.9C2^3"); // GroupNames label
G:=SmallGroup(192,1307);
// by ID
G=gap.SmallGroup(192,1307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^24=b^2=1,c^2=d^2=a^12,b*a*b=a^11,a*c=c*a,d*a*d^-1=a^13,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations