metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.11C23, C4⋊F5⋊3C2, (C2×C4)⋊4F5, (C2×C20)⋊4C4, (C4×D5)⋊6C4, (C4×F5)⋊4C2, C22⋊F5.C2, C5⋊(C42⋊C2), C4.13(C2×F5), C20.21(C2×C4), (C2×Dic5)⋊9C4, C2.7(C22×F5), C22.7(C2×F5), D5.1(C4○D4), D10.17(C2×C4), C10.6(C22×C4), (C2×F5).2C22, Dic5.17(C2×C4), (C4×D5).36C22, (C22×D5).38C22, (C2×C4×D5).16C2, (C2×C10).18(C2×C4), SmallGroup(160,205)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.C23
G = < a,b,c,d,e | a10=b2=e2=1, c2=a-1b, d2=a5, bab=a-1, cac-1=a3, ad=da, ae=ea, cbc-1=a2b, bd=db, be=eb, cd=dc, ece=a5c, de=ed >
Subgroups: 244 in 76 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, D10.C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, F5, C42⋊C2, C2×F5, C22×F5, D10.C23
Character table of D10.C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -1 | -i | i | i | -i | i | -i | -i | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ10 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -1 | i | -i | i | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -i | 1 | -i | i | i | -i | -i | i | i | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | 1 | i | -i | i | -i | -i | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | i | 1 | i | -i | -i | i | i | -i | -i | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | 1 | -i | i | -i | i | i | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -1 | i | -i | -i | i | -i | i | i | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -1 | -i | i | -i | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ17 | 2 | -2 | 0 | 2 | -2 | 0 | -2i | 2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 0 | 2 | -2 | 0 | 2i | -2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 0 | -2 | 2 | 0 | -2i | 2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 0 | -2 | 2 | 0 | 2i | -2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from C2×F5 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | 1 | √5 | i | -√-5 | √-5 | -i | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | 1 | √5 | -i | √-5 | -√-5 | i | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | √5 | 1 | -√5 | i | √-5 | -√-5 | -i | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | √5 | 1 | -√5 | -i | -√-5 | √-5 | i | complex faithful |
►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 5)(2 4)(6 10)(7 9)(11 19)(12 18)(13 17)(14 16)(21 29)(22 28)(23 27)(24 26)(31 39)(32 38)(33 37)(34 36)
(1 13 6 18)(2 20 5 11)(3 17 4 14)(7 15 10 16)(8 12 9 19)(21 39 30 32)(22 36 29 35)(23 33 28 38)(24 40 27 31)(25 37 26 34)
(1 28 6 23)(2 29 7 24)(3 30 8 25)(4 21 9 26)(5 22 10 27)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
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), (1,5)(2,4)(6,10)(7,9)(11,19)(12,18)(13,17)(14,16)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36), (1,13,6,18)(2,20,5,11)(3,17,4,14)(7,15,10,16)(8,12,9,19)(21,39,30,32)(22,36,29,35)(23,33,28,38)(24,40,27,31)(25,37,26,34), (1,28,6,23)(2,29,7,24)(3,30,8,25)(4,21,9,26)(5,22,10,27)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,11)(10,12)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)>;
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), (1,5)(2,4)(6,10)(7,9)(11,19)(12,18)(13,17)(14,16)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36), (1,13,6,18)(2,20,5,11)(3,17,4,14)(7,15,10,16)(8,12,9,19)(21,39,30,32)(22,36,29,35)(23,33,28,38)(24,40,27,31)(25,37,26,34), (1,28,6,23)(2,29,7,24)(3,30,8,25)(4,21,9,26)(5,22,10,27)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,11)(10,12)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40) );
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)], [(1,5),(2,4),(6,10),(7,9),(11,19),(12,18),(13,17),(14,16),(21,29),(22,28),(23,27),(24,26),(31,39),(32,38),(33,37),(34,36)], [(1,13,6,18),(2,20,5,11),(3,17,4,14),(7,15,10,16),(8,12,9,19),(21,39,30,32),(22,36,29,35),(23,33,28,38),(24,40,27,31),(25,37,26,34)], [(1,28,6,23),(2,29,7,24),(3,30,8,25),(4,21,9,26),(5,22,10,27),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,11),(10,12),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)]])
D10.C23 is a maximal subgroup of
C42⋊6F5 C42⋊3F5 (C2×C8)⋊F5 C20.24C42 M4(2)⋊3F5 M4(2)⋊4F5 C20.12C42 (C2×C8)⋊6F5 M4(2)⋊1F5 M4(2)⋊5F5 C23⋊F5⋊5C2 (D4×C10)⋊C4 (C2×D4)⋊6F5 (C2×D4)⋊8F5 (C2×Q8)⋊4F5 (C2×Q8)⋊6F5 (C2×Q8)⋊7F5 C4○D20⋊C4 D4⋊F5⋊C2 D10.C24 D5.2- 1+4 C4○D4×F5 D5.2+ 1+4 C4⋊F5⋊3S3 (C4×S3)⋊F5 C22⋊F5.S3 (C2×C12)⋊6F5
D10.C23 is a maximal quotient of
C42.6F5 C42.12F5 C42.15F5 C42.7F5 C42⋊4F5 C4×C4⋊F5 C42⋊9F5 C42⋊5F5 C23.(C2×F5) C10.(C4×D4) C4⋊C4.7F5 C4⋊C4⋊5F5 Dic5.12M4(2) C20.34M4(2) Dic5.13M4(2) C20.30M4(2) C4×C22⋊F5 (C22×C4)⋊7F5 D10⋊6(C4⋊C4) C4⋊F5⋊3S3 (C4×S3)⋊F5 C22⋊F5.S3 (C2×C12)⋊6F5
Matrix representation of D10.C23 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,0,0,1,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D10.C23 in GAP, Magma, Sage, TeX
D_{10}.C_2^3 % in TeX
G:=Group("D10.C2^3"); // GroupNames label
G:=SmallGroup(160,205);
// by ID
G=gap.SmallGroup(160,205);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,362,2309,599]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=e^2=1,c^2=a^-1*b,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,a*e=e*a,c*b*c^-1=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^5*c,d*e=e*d>;
// generators/relations
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