metabelian, supersoluble, monomial
Aliases: C102.8C4, C52⋊14M4(2), (C2×C10).8F5, C52⋊5C8⋊6C2, C10.26(C2×F5), C5⋊3(C22.F5), C22.(C52⋊C4), C52⋊6C4.10C4, C52⋊6C4.25C22, C2.6(C2×C52⋊C4), (C5×C10).39(C2×C4), (C2×C52⋊6C4).11C2, SmallGroup(400,161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊14M4(2)
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a2, ad=da, cbc-1=b3, bd=db, dcd=c5 >
Subgroups: 316 in 56 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2, C4, C22, C5, C5, C8, C2×C4, C10, C10, M4(2), Dic5, C2×C10, C2×C10, C52, C5⋊C8, C2×Dic5, C5×C10, C5×C10, C22.F5, C52⋊6C4, C102, C52⋊5C8, C2×C52⋊6C4, C52⋊14M4(2)
Quotients: C1, C2, C4, C22, C2×C4, M4(2), F5, C2×F5, C22.F5, C52⋊C4, C2×C52⋊C4, C52⋊14M4(2)
►On 40 points
(1 37 10 26 18)(2 11 19 38 27)(3 20 28 12 39)(4 29 40 21 13)(5 33 14 30 22)(6 15 23 34 31)(7 24 32 16 35)(8 25 36 17 9)
(1 26 37 18 10)(2 19 27 11 38)(3 12 20 39 28)(4 40 13 29 21)(5 30 33 22 14)(6 23 31 15 34)(7 16 24 35 32)(8 36 9 25 17)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)
G:=sub<Sym(40)| (1,37,10,26,18)(2,11,19,38,27)(3,20,28,12,39)(4,29,40,21,13)(5,33,14,30,22)(6,15,23,34,31)(7,24,32,16,35)(8,25,36,17,9), (1,26,37,18,10)(2,19,27,11,38)(3,12,20,39,28)(4,40,13,29,21)(5,30,33,22,14)(6,23,31,15,34)(7,16,24,35,32)(8,36,9,25,17), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)>;
G:=Group( (1,37,10,26,18)(2,11,19,38,27)(3,20,28,12,39)(4,29,40,21,13)(5,33,14,30,22)(6,15,23,34,31)(7,24,32,16,35)(8,25,36,17,9), (1,26,37,18,10)(2,19,27,11,38)(3,12,20,39,28)(4,40,13,29,21)(5,30,33,22,14)(6,23,31,15,34)(7,16,24,35,32)(8,36,9,25,17), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40) );
G=PermutationGroup([[(1,37,10,26,18),(2,11,19,38,27),(3,20,28,12,39),(4,29,40,21,13),(5,33,14,30,22),(6,15,23,34,31),(7,24,32,16,35),(8,25,36,17,9)], [(1,26,37,18,10),(2,19,27,11,38),(3,12,20,39,28),(4,40,13,29,21),(5,30,33,22,14),(6,23,31,15,34),(7,16,24,35,32),(8,36,9,25,17)], [(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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | ··· | 5F | 8A | 8B | 8C | 8D | 10A | ··· | 10R |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 |
| size | 1 | 1 | 2 | 25 | 25 | 50 | 4 | ··· | 4 | 50 | 50 | 50 | 50 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | F5 | C2×F5 | C22.F5 | C52⋊C4 | C2×C52⋊C4 | C52⋊14M4(2) |
| kernel | C52⋊14M4(2) | C52⋊5C8 | C2×C52⋊6C4 | C52⋊6C4 | C102 | C52 | C2×C10 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C52⋊14M4(2) ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 6 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 33 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 6 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 7 | 35 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,0,0,0,0,6,0,0,0,0,0,0,0,40,33,0,0,0,0,1,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,6,0],[0,32,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,8,0,0,0,0,35,34,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C52⋊14M4(2) in GAP, Magma, Sage, TeX
C_5^2\rtimes_{14}M_4(2) % in TeX
G:=Group("C5^2:14M4(2)"); // GroupNames label
G:=SmallGroup(400,161);
// by ID
G=gap.SmallGroup(400,161);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,1444,496,5765,2897]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^2,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d=c^5>;
// generators/relations