metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.25D12, C24.50D4, M5(2)⋊3S3, C12.6M4(2), (C22×S3).C8, (C2×Dic3).C8, (C2×C8).152D6, C3⋊1(C23.C8), C22.5(S3×C8), C2.10(D6⋊C8), C4.42(D6⋊C4), C8.46(C3⋊D4), C6.9(C22⋊C8), (C3×M5(2))⋊7C2, C12.C8⋊10C2, C4.10(C8⋊S3), C12.57(C22⋊C4), (C2×C24).263C22, (C2×C3⋊C8).2C4, (S3×C2×C4).1C4, (C2×C6).3(C2×C8), (C2×C4).137(C4×S3), (C2×C12).52(C2×C4), (C2×C8⋊S3).10C2, SmallGroup(192,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.25D12
G = < a,b,c | a8=1, b12=a6, c2=a, bab-1=a5, ac=ca, cbc-1=a3b11 >
Subgroups: 152 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C16, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, M5(2), M5(2), C2×M4(2), C3⋊C16, C48, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C23.C8, C12.C8, C3×M5(2), C2×C8⋊S3, C8.25D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), C4×S3, D12, C3⋊D4, C22⋊C8, S3×C8, C8⋊S3, D6⋊C4, C23.C8, D6⋊C8, C8.25D12
►On 48 points
(1 19 37 7 25 43 13 31)(2 44 38 32 26 20 14 8)(3 21 39 9 27 45 15 33)(4 46 40 34 28 22 16 10)(5 23 41 11 29 47 17 35)(6 48 42 36 30 24 18 12)
(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 42 19 36 37 30 7 24 25 18 43 12 13 6 31 48)(2 11 44 29 38 47 32 17 26 35 20 5 14 23 8 41)(3 4 21 46 39 40 9 34 27 28 45 22 15 16 33 10)
G:=sub<Sym(48)| (1,19,37,7,25,43,13,31)(2,44,38,32,26,20,14,8)(3,21,39,9,27,45,15,33)(4,46,40,34,28,22,16,10)(5,23,41,11,29,47,17,35)(6,48,42,36,30,24,18,12), (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), (1,42,19,36,37,30,7,24,25,18,43,12,13,6,31,48)(2,11,44,29,38,47,32,17,26,35,20,5,14,23,8,41)(3,4,21,46,39,40,9,34,27,28,45,22,15,16,33,10)>;
G:=Group( (1,19,37,7,25,43,13,31)(2,44,38,32,26,20,14,8)(3,21,39,9,27,45,15,33)(4,46,40,34,28,22,16,10)(5,23,41,11,29,47,17,35)(6,48,42,36,30,24,18,12), (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), (1,42,19,36,37,30,7,24,25,18,43,12,13,6,31,48)(2,11,44,29,38,47,32,17,26,35,20,5,14,23,8,41)(3,4,21,46,39,40,9,34,27,28,45,22,15,16,33,10) );
G=PermutationGroup([[(1,19,37,7,25,43,13,31),(2,44,38,32,26,20,14,8),(3,21,39,9,27,45,15,33),(4,46,40,34,28,22,16,10),(5,23,41,11,29,47,17,35),(6,48,42,36,30,24,18,12)], [(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)], [(1,42,19,36,37,30,7,24,25,18,43,12,13,6,31,48),(2,11,44,29,38,47,32,17,26,35,20,5,14,23,8,41),(3,4,21,46,39,40,9,34,27,28,45,22,15,16,33,10)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 12 | 2 | 1 | 1 | 2 | 12 | 2 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | C8⋊S3 | S3×C8 | C23.C8 | C8.25D12 |
| kernel | C8.25D12 | C12.C8 | C3×M5(2) | C2×C8⋊S3 | C2×C3⋊C8 | S3×C2×C4 | C2×Dic3 | C22×S3 | M5(2) | C24 | C2×C8 | C12 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C8.25D12 ►in GL4(𝔽5) generated by
| 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 1 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 2 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 3 | 0 | 1 |
| 4 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [0,0,4,0,0,4,0,2,3,0,0,0,0,3,0,1],[0,0,0,1,0,0,4,0,0,3,0,1,4,0,2,0],[0,0,0,4,1,0,3,0,0,3,0,1,0,0,1,0] >;
C8.25D12 in GAP, Magma, Sage, TeX
C_8._{25}D_{12} % in TeX
G:=Group("C8.25D12"); // GroupNames label
G:=SmallGroup(192,73);
// by ID
G=gap.SmallGroup(192,73);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,100,570,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^12=a^6,c^2=a,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^3*b^11>;
// generators/relations