metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.1D10, C22.2D20, C5⋊3(C23⋊C4), C22⋊C4⋊1D5, (C2×Dic5)⋊1C4, (C2×C10).27D4, (C22×D5)⋊1C4, C23.D5⋊1C2, C22.3(C4×D5), C22.8(C5⋊D4), C2.4(D10⋊C4), C10.13(C22⋊C4), (C22×C10).5C22, (C5×C22⋊C4)⋊1C2, (C2×C5⋊D4).1C2, (C2×C10).21(C2×C4), SmallGroup(160,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.1D10
G = < a,b,c,d | a2=b2=c20=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Smallest permutation representation of C23.1D10
►On 40 points
(1 30)(3 32)(5 34)(7 36)(9 38)(11 40)(13 22)(15 24)(17 26)(19 28)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)
(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 24 30 15)(2 23)(3 13 32 22)(4 12)(5 40 34 11)(6 39)(7 9 36 38)(10 35)(14 31)(16 20)(17 28 26 19)(18 27)(21 33)(25 29)
G:=sub<Sym(40)| (1,30)(3,32)(5,34)(7,36)(9,38)(11,40)(13,22)(15,24)(17,26)(19,28), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29), (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), (1,24,30,15)(2,23)(3,13,32,22)(4,12)(5,40,34,11)(6,39)(7,9,36,38)(10,35)(14,31)(16,20)(17,28,26,19)(18,27)(21,33)(25,29)>;
G:=Group( (1,30)(3,32)(5,34)(7,36)(9,38)(11,40)(13,22)(15,24)(17,26)(19,28), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29), (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), (1,24,30,15)(2,23)(3,13,32,22)(4,12)(5,40,34,11)(6,39)(7,9,36,38)(10,35)(14,31)(16,20)(17,28,26,19)(18,27)(21,33)(25,29) );
G=PermutationGroup([[(1,30),(3,32),(5,34),(7,36),(9,38),(11,40),(13,22),(15,24),(17,26),(19,28)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29)], [(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)], [(1,24,30,15),(2,23),(3,13,32,22),(4,12),(5,40,34,11),(6,39),(7,9,36,38),(10,35),(14,31),(16,20),(17,28,26,19),(18,27),(21,33),(25,29)]])
C23.1D10 is a maximal subgroup of
C23⋊C4⋊5D5 C23⋊D20 C23.5D20 D5×C23⋊C4 (C2×D20)⋊25C4 C24⋊2D10 C22⋊C4⋊D10 (C2×C6).D20 C15⋊9(C23⋊C4) C23.6D30
C23.1D10 is a maximal quotient of
(C2×D20)⋊C4 C4⋊Dic5⋊C4 C23.30D20 C5⋊3(C23⋊C8) (C2×Dic5)⋊C8 C22.2D40 C5⋊3C2≀C4 (C2×C20).D4 C23.D20 C23.2D20 C23.3D20 C23.4D20 (C2×C4).D20 (C2×Q8).D10 C24.2D10 (C2×C6).D20 C15⋊9(C23⋊C4) C23.6D30
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 4 | 4 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | C23.1D10 |
| kernel | C23.1D10 | C23.D5 | C5×C22⋊C4 | C2×C5⋊D4 | C2×Dic5 | C22×D5 | C2×C10 | C22⋊C4 | C23 | C22 | C22 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 4 |
Matrix representation of C23.1D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 6 |
| 35 | 18 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 23 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,0,35,23,0,0,18,20,0,40,0,0,1,6,0,0],[35,23,0,0,18,6,0,0,0,0,0,1,0,0,1,0] >;
C23.1D10 in GAP, Magma, Sage, TeX
C_2^3._1D_{10} % in TeX
G:=Group("C2^3.1D10"); // GroupNames label
G:=SmallGroup(160,13);
// by ID
G=gap.SmallGroup(160,13);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,297,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
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