direct product, metabelian, supersoluble, monomial
Aliases: C5×C23.D5, C102⋊10C4, C102.20C22, (C2×C10)⋊4C20, C23.(C5×D5), C22⋊(C5×Dic5), (C2×C10)⋊3Dic5, (C5×C10).32D4, C10.11(C5×D4), C10.16(C2×C20), (C10×Dic5)⋊4C2, (C2×Dic5)⋊2C10, (C2×C10).44D10, (C2×C102).2C2, C2.5(C10×Dic5), C22.7(D5×C10), (C22×C10).1D5, C10.33(C5⋊D4), C52⋊12(C22⋊C4), (C22×C10).4C10, C10.28(C2×Dic5), C5⋊3(C5×C22⋊C4), C2.3(C5×C5⋊D4), (C2×C10).9(C2×C10), (C5×C10).63(C2×C4), SmallGroup(400,91)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.D5
G = < a,b,c,d,e,f | a5=b2=c2=d2=e5=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 212 in 100 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C5, C2×C4, C23, C10, C10, C10, C22⋊C4, Dic5, C20, C2×C10, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C22×C10, C22×C10, C5×C10, C5×C10, C5×C10, C23.D5, C5×C22⋊C4, C5×Dic5, C102, C102, C102, C10×Dic5, C2×C102, C5×C23.D5
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, D5, C10, C22⋊C4, Dic5, C20, D10, C2×C10, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D5, C23.D5, C5×C22⋊C4, C5×Dic5, D5×C10, C10×Dic5, C5×C5⋊D4, C5×C23.D5
►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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 25 24 23 22)(26 30 29 28 27)(31 35 34 33 32)(36 40 39 38 37)
(1 32 11 22)(2 33 12 23)(3 34 13 24)(4 35 14 25)(5 31 15 21)(6 37 16 27)(7 38 17 28)(8 39 18 29)(9 40 19 30)(10 36 20 26)
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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)(36,40,39,38,37), (1,32,11,22)(2,33,12,23)(3,34,13,24)(4,35,14,25)(5,31,15,21)(6,37,16,27)(7,38,17,28)(8,39,18,29)(9,40,19,30)(10,36,20,26)>;
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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)(36,40,39,38,37), (1,32,11,22)(2,33,12,23)(3,34,13,24)(4,35,14,25)(5,31,15,21)(6,37,16,27)(7,38,17,28)(8,39,18,29)(9,40,19,30)(10,36,20,26) );
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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,25,24,23,22),(26,30,29,28,27),(31,35,34,33,32),(36,40,39,38,37)], [(1,32,11,22),(2,33,12,23),(3,34,13,24),(4,35,14,25),(5,31,15,21),(6,37,16,27),(7,38,17,28),(8,39,18,29),(9,40,19,30),(10,36,20,26)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | ··· | 10L | 10M | ··· | 10CL | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | D5 | Dic5 | D10 | C5⋊D4 | C5×D4 | C5×D5 | C5×Dic5 | D5×C10 | C5×C5⋊D4 |
| kernel | C5×C23.D5 | C10×Dic5 | C2×C102 | C102 | C23.D5 | C2×Dic5 | C22×C10 | C2×C10 | C5×C10 | C22×C10 | C2×C10 | C2×C10 | C10 | C10 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 2 | 2 | 4 | 2 | 8 | 8 | 8 | 16 | 8 | 32 |
Matrix representation of C5×C23.D5 ►in GL3(𝔽41) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 1 |
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 37 |
| 9 | 0 | 0 |
| 0 | 0 | 37 |
| 0 | 31 | 0 |
G:=sub<GL(3,GF(41))| [10,0,0,0,10,0,0,0,10],[1,0,0,0,40,0,0,0,1],[40,0,0,0,40,0,0,0,40],[1,0,0,0,40,0,0,0,40],[1,0,0,0,10,0,0,0,37],[9,0,0,0,0,31,0,37,0] >;
C5×C23.D5 in GAP, Magma, Sage, TeX
C_5\times C_2^3.D_5
% in TeX
G:=Group("C5xC2^3.D5"); // GroupNames label
G:=SmallGroup(400,91);
// by ID
G=gap.SmallGroup(400,91);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,11525]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^5=1,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations