metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10:5D4, C23:1D10, C5:2C22wrC2, (C2xD4):3D5, (C2xC10):2D4, (C2xC4):2D10, (D4xC10):8C2, C2.25(D4xD5), (C2xC20):7C22, C10.49(C2xD4), (C23xD5):2C2, C22:2(C5:D4), C23.D5:10C2, D10:C4:14C2, (C2xC10).52C23, (C22xC10):3C22, (C2xDic5):2C22, C22.59(C22xD5), (C22xD5).28C22, (C2xC5:D4):4C2, C2.13(C2xC5:D4), SmallGroup(160,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23:D10
G = < a,b,c,d,e | a2=b2=c2=d10=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 496 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, C23, C23, D5, C10, C10, C10, C22:C4, C2xD4, C2xD4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC10, C22wrC2, C2xDic5, C5:D4, C2xC20, C5xD4, C22xD5, C22xD5, C22xC10, D10:C4, C23.D5, C2xC5:D4, D4xC10, C23xD5, C23:D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C22wrC2, C5:D4, C22xD5, D4xD5, C2xC5:D4, C23:D10
►On 40 points
(1 34)(2 12)(3 36)(4 14)(5 38)(6 16)(7 40)(8 18)(9 32)(10 20)(11 28)(13 30)(15 22)(17 24)(19 26)(21 37)(23 39)(25 31)(27 33)(29 35)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 31)(19 32)(20 33)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 30)(9 29)(10 28)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 40)(20 39)
G:=sub<Sym(40)| (1,34)(2,12)(3,36)(4,14)(5,38)(6,16)(7,40)(8,18)(9,32)(10,20)(11,28)(13,30)(15,22)(17,24)(19,26)(21,37)(23,39)(25,31)(27,33)(29,35), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,31)(19,32)(20,33), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,30)(9,29)(10,28)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,40)(20,39)>;
G:=Group( (1,34)(2,12)(3,36)(4,14)(5,38)(6,16)(7,40)(8,18)(9,32)(10,20)(11,28)(13,30)(15,22)(17,24)(19,26)(21,37)(23,39)(25,31)(27,33)(29,35), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,31)(19,32)(20,33), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,30)(9,29)(10,28)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,40)(20,39) );
G=PermutationGroup([[(1,34),(2,12),(3,36),(4,14),(5,38),(6,16),(7,40),(8,18),(9,32),(10,20),(11,28),(13,30),(15,22),(17,24),(19,26),(21,37),(23,39),(25,31),(27,33),(29,35)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,16),(12,17),(13,18),(14,19),(15,20),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,28),(2,29),(3,30),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,31),(19,32),(20,33)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,30),(9,29),(10,28),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,40),(20,39)]])
C23:D10 is a maximal subgroup of
C5:3C2wrC4 C23.3D20 C23:D20 2+ 1+4:2D5 C42:12D10 D20:23D4 C42:16D10 C42:17D10 D5xC22wrC2 C24:3D10 C24:4D10 C24.33D10 C24.34D10 C24:5D10 C10.372+ 1+4 C4:C4:21D10 C10.382+ 1+4 D20:19D4 C10.402+ 1+4 D20:20D4 C10.422+ 1+4 C10.462+ 1+4 C10.482+ 1+4 C10.1202+ 1+4 C4:C4:28D10 C10.612+ 1+4 C10.1222+ 1+4 C10.622+ 1+4 C10.682+ 1+4 C42:20D10 C42:21D10 C42:22D10 C42:26D10 D20:11D4 C42:28D10 D4xC5:D4 C24:8D10 (C2xC20):15D4 C10.1452+ 1+4 C10.1462+ 1+4 D30:4D4 (C2xC30):D4 D30:19D4 D30:8D4 D30:17D4 D10:S4
C23:D10 is a maximal quotient of
C24.46D10 C23:Dic10 C24.48D10 C24.12D10 C24.14D10 C23:2D20 (C2xC4):Dic10 D10:5(C4:C4) (C2xC4):3D20 C24:2D10 D20:16D4 D20:17D4 Dic10:17D4 D20.36D4 D20.37D4 Dic10.37D4 C22:C4:D10 C42:5D10 D20.14D4 D20:5D4 D20.15D4 D20:D4 Dic10:D4 D10:6SD16 D10:8SD16 D20:7D4 Dic10.16D4 D10:5Q16 D20.17D4 D20:18D4 D20.38D4 D20.39D4 D20.40D4 C24.18D10 C24.21D10 D30:4D4 (C2xC30):D4 D30:19D4 D30:8D4 D30:17D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 4 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5:D4 | D4xD5 |
| kernel | C23:D10 | D10:C4 | C23.D5 | C2xC5:D4 | D4xC10 | C23xD5 | D10 | C2xC10 | C2xD4 | C2xC4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 4 |
Matrix representation of C23:D10 ►in GL4(F41) generated by
| 17 | 6 | 0 | 0 |
| 34 | 24 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 6 | 0 | 0 |
| 34 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 6 | 0 | 0 |
| 33 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [17,34,0,0,6,24,0,0,0,0,0,1,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,34,0,0,6,7,0,0,0,0,40,0,0,0,0,1],[34,33,0,0,6,7,0,0,0,0,1,0,0,0,0,40] >;
C23:D10 in GAP, Magma, Sage, TeX
C_2^3\rtimes D_{10} % in TeX
G:=Group("C2^3:D10"); // GroupNames label
G:=SmallGroup(160,158);
// by ID
G=gap.SmallGroup(160,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^10=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations