metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.46D4, C4.11D20, M4(2)⋊3D5, (C2×C4).1D10, (C2×D20).7C2, C5⋊2(C4.D4), C4.Dic5⋊2C2, C22.4(C4×D5), C4.21(C5⋊D4), (C5×M4(2))⋊7C2, (C22×D5).1C4, (C2×C20).13C22, C2.9(D10⋊C4), C10.19(C22⋊C4), (C2×C10).22(C2×C4), SmallGroup(160,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.46D4
G = < a,b,c | a20=c2=1, b4=a10, bab-1=cac=a-1, cbc=a15b3 >
Smallest permutation representation of C20.46D4
►On 40 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)
(1 36 16 21 11 26 6 31)(2 35 17 40 12 25 7 30)(3 34 18 39 13 24 8 29)(4 33 19 38 14 23 9 28)(5 32 20 37 15 22 10 27)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 20)(18 19)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)
G:=sub<Sym(40)| (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), (1,36,16,21,11,26,6,31)(2,35,17,40,12,25,7,30)(3,34,18,39,13,24,8,29)(4,33,19,38,14,23,9,28)(5,32,20,37,15,22,10,27), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,20)(18,19)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)>;
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), (1,36,16,21,11,26,6,31)(2,35,17,40,12,25,7,30)(3,34,18,39,13,24,8,29)(4,33,19,38,14,23,9,28)(5,32,20,37,15,22,10,27), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,20)(18,19)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34) );
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)], [(1,36,16,21,11,26,6,31),(2,35,17,40,12,25,7,30),(3,34,18,39,13,24,8,29),(4,33,19,38,14,23,9,28),(5,32,20,37,15,22,10,27)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,20),(18,19),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34)]])
C20.46D4 is a maximal subgroup of
D5×C4.D4 D20.3D4 M4(2).21D10 D20.6D4 D4⋊4D20 M4(2)⋊D10 C8.21D20 C8.24D20 M4(2).31D10 D4.3D20 D4.4D20 D20⋊18D4 M4(2).D10 D20.39D4 M4(2).15D10 C60.28D4 C60.29D4 M4(2)⋊D15
C20.46D4 is a maximal quotient of
C42.D10 C5⋊3(C23⋊C8) C4.Dic20 C4.D40 M4(2)⋊Dic5 C60.28D4 C60.29D4 M4(2)⋊D15
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D5 | D10 | D20 | C5⋊D4 | C4×D5 | C4.D4 | C20.46D4 |
| kernel | C20.46D4 | C4.Dic5 | C5×M4(2) | C2×D20 | C22×D5 | C20 | M4(2) | C2×C4 | C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 4 |
Matrix representation of C20.46D4 ►in GL4(𝔽41) generated by
| 14 | 30 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 14 | 30 |
| 0 | 0 | 11 | 9 |
| 30 | 32 | 19 | 23 |
| 27 | 11 | 13 | 22 |
| 5 | 25 | 11 | 9 |
| 31 | 36 | 14 | 30 |
| 11 | 9 | 0 | 0 |
| 14 | 30 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 14 | 30 |
G:=sub<GL(4,GF(41))| [14,11,0,0,30,9,0,0,0,0,14,11,0,0,30,9],[30,27,5,31,32,11,25,36,19,13,11,14,23,22,9,30],[11,14,0,0,9,30,0,0,0,0,11,14,0,0,9,30] >;
C20.46D4 in GAP, Magma, Sage, TeX
C_{20}._{46}D_4 % in TeX
G:=Group("C20.46D4"); // GroupNames label
G:=SmallGroup(160,30);
// by ID
G=gap.SmallGroup(160,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,86,297,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=c^2=1,b^4=a^10,b*a*b^-1=c*a*c=a^-1,c*b*c=a^15*b^3>;
// generators/relations
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