metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.52D4, C12.25M4(2), (C2×C12).1C8, (C2×C24).2C4, (C2×C8).153D6, C3⋊2(C23.C8), C23.2(C3⋊C8), (C22×C6).2C8, (C2×C8).2Dic3, C8.33(C3⋊D4), (C22×C12).7C4, C12.C8⋊11C2, C6.18(C22⋊C8), (C2×M4(2)).4S3, (C6×M4(2)).6C2, C4.7(C4.Dic3), (C22×C4).7Dic3, C12.93(C22⋊C4), (C2×C24).265C22, C4.26(C6.D4), C2.7(C12.55D4), (C2×C4).(C3⋊C8), C22.4(C2×C3⋊C8), (C2×C6).32(C2×C8), (C2×C12).303(C2×C4), (C2×C4).75(C2×Dic3), SmallGroup(192,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.D4
G = < a,b,c | a24=1, b4=a18, c2=a9, bab-1=a5, cac-1=a17, cbc-1=a15b3 >
Subgroups: 104 in 58 conjugacy classes, 31 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C16, C2×C8, M4(2), C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, M5(2), C2×M4(2), C3⋊C16, C2×C24, C3×M4(2), C22×C12, C23.C8, C12.C8, C6×M4(2), C24.D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C2×C3⋊C8, C4.Dic3, C6.D4, C23.C8, C12.55D4, C24.D4
►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)
(1 46 10 43 19 40 4 37 13 34 22 31 7 28 16 25)(2 27 11 48 20 45 5 42 14 39 23 36 8 33 17 30)(3 32 12 29 21 26 6 47 15 44 24 41 9 38 18 35)
(1 34 10 43 19 28 4 37 13 46 22 31 7 40 16 25)(2 27 11 36 20 45 5 30 14 39 23 48 8 33 17 42)(3 44 12 29 21 38 6 47 15 32 24 41 9 26 18 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), (1,46,10,43,19,40,4,37,13,34,22,31,7,28,16,25)(2,27,11,48,20,45,5,42,14,39,23,36,8,33,17,30)(3,32,12,29,21,26,6,47,15,44,24,41,9,38,18,35), (1,34,10,43,19,28,4,37,13,46,22,31,7,40,16,25)(2,27,11,36,20,45,5,30,14,39,23,48,8,33,17,42)(3,44,12,29,21,38,6,47,15,32,24,41,9,26,18,35)>;
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), (1,46,10,43,19,40,4,37,13,34,22,31,7,28,16,25)(2,27,11,48,20,45,5,42,14,39,23,36,8,33,17,30)(3,32,12,29,21,26,6,47,15,44,24,41,9,38,18,35), (1,34,10,43,19,28,4,37,13,46,22,31,7,40,16,25)(2,27,11,36,20,45,5,30,14,39,23,48,8,33,17,42)(3,44,12,29,21,38,6,47,15,32,24,41,9,26,18,35) );
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)], [(1,46,10,43,19,40,4,37,13,34,22,31,7,28,16,25),(2,27,11,48,20,45,5,42,14,39,23,36,8,33,17,30),(3,32,12,29,21,26,6,47,15,44,24,41,9,38,18,35)], [(1,34,10,43,19,28,4,37,13,46,22,31,7,40,16,25),(2,27,11,36,20,45,5,30,14,39,23,48,8,33,17,42),(3,44,12,29,21,38,6,47,15,32,24,41,9,26,18,35)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | ··· | 12 | 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 | C4 | C4 | C8 | C8 | S3 | D4 | Dic3 | D6 | Dic3 | M4(2) | C3⋊D4 | C3⋊C8 | C3⋊C8 | C4.Dic3 | C23.C8 | C24.D4 |
| kernel | C24.D4 | C12.C8 | C6×M4(2) | C2×C24 | C22×C12 | C2×C12 | C22×C6 | C2×M4(2) | C24 | C2×C8 | C2×C8 | C22×C4 | C12 | C8 | C2×C4 | C23 | C4 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C24.D4 ►in GL4(𝔽97) generated by
| 61 | 10 | 0 | 0 |
| 65 | 36 | 0 | 0 |
| 0 | 0 | 35 | 28 |
| 0 | 0 | 85 | 62 |
| 0 | 0 | 1 | 22 |
| 0 | 0 | 0 | 96 |
| 96 | 37 | 0 | 0 |
| 53 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 59 | 0 | 0 |
| 44 | 96 | 0 | 0 |
G:=sub<GL(4,GF(97))| [61,65,0,0,10,36,0,0,0,0,35,85,0,0,28,62],[0,0,96,53,0,0,37,1,1,0,0,0,22,96,0,0],[0,0,1,44,0,0,59,96,1,0,0,0,0,1,0,0] >;
C24.D4 in GAP, Magma, Sage, TeX
C_{24}.D_4 % in TeX
G:=Group("C24.D4"); // GroupNames label
G:=SmallGroup(192,112);
// by ID
G=gap.SmallGroup(192,112);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,100,1123,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^5,c*a*c^-1=a^17,c*b*c^-1=a^15*b^3>;
// generators/relations