metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23:D20, (C2xC4):D20, C5:1C2wrC22, (C2xC20):1D4, C23:C4:3D5, C22:C4:1D10, (C2xDic5):1D4, (C22xD5):1D4, (C22xC10):2D4, C23:D10:1C2, D4:6D10:1C2, C22:D20:1C2, (C2xD4).11D10, C22.8(C2xD20), C22.24(D4xD5), C10.13C22wrC2, (D4xC10).8C22, (C23xD5):1C22, C23.D5:1C22, C23.2(C22xD5), C23.1D10:1C2, (C22xC10).2C23, C2.16(C22:D20), (C5xC23:C4):4C2, (C2xC10).17(C2xD4), (C5xC22:C4):1C22, (C2xC5:D4).2C22, SmallGroup(320,368)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23:D20
G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, ac=ca, dad-1=eae=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1150 in 198 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, Q8, C23, C23, D5, C10, C10, C22:C4, C22:C4, C2xD4, C2xD4, C4oD4, C24, Dic5, C20, D10, C2xC10, C2xC10, C2xC10, C23:C4, C23:C4, C22wrC2, 2+ 1+4, Dic10, C4xD5, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xC10, C2wrC22, D10:C4, C23.D5, C5xC22:C4, C2xD20, C4oD20, D4xD5, D4:2D5, C2xC5:D4, C2xC5:D4, D4xC10, C23xD5, C23.1D10, C5xC23:C4, C22:D20, C23:D10, D4:6D10, C23:D20
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C22wrC2, D20, C22xD5, C2wrC22, C2xD20, D4xD5, C22:D20, C23:D20
(1 34)(2 30)(3 26)(4 22)(5 38)(6 28)(7 24)(8 40)(9 36)(10 32)(11 35)(12 21)(13 27)(14 33)(15 39)(16 25)(17 31)(18 37)(19 23)(20 29)
(1 15)(2 11)(3 17)(4 13)(5 19)(6 14)(7 20)(8 16)(9 12)(10 18)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 16)(12 17)(13 18)(14 19)(15 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 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 2)(3 5)(6 9)(7 8)(11 15)(12 14)(16 20)(17 19)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)
G:=sub<Sym(40)| (1,34)(2,30)(3,26)(4,22)(5,38)(6,28)(7,24)(8,40)(9,36)(10,32)(11,35)(12,21)(13,27)(14,33)(15,39)(16,25)(17,31)(18,37)(19,23)(20,29), (1,15)(2,11)(3,17)(4,13)(5,19)(6,14)(7,20)(8,16)(9,12)(10,18)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39), (1,7)(2,8)(3,9)(4,10)(5,6)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,2)(3,5)(6,9)(7,8)(11,15)(12,14)(16,20)(17,19)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)>;
G:=Group( (1,34)(2,30)(3,26)(4,22)(5,38)(6,28)(7,24)(8,40)(9,36)(10,32)(11,35)(12,21)(13,27)(14,33)(15,39)(16,25)(17,31)(18,37)(19,23)(20,29), (1,15)(2,11)(3,17)(4,13)(5,19)(6,14)(7,20)(8,16)(9,12)(10,18)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39), (1,7)(2,8)(3,9)(4,10)(5,6)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,2)(3,5)(6,9)(7,8)(11,15)(12,14)(16,20)(17,19)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40) );
G=PermutationGroup([[(1,34),(2,30),(3,26),(4,22),(5,38),(6,28),(7,24),(8,40),(9,36),(10,32),(11,35),(12,21),(13,27),(14,33),(15,39),(16,25),(17,31),(18,37),(19,23),(20,29)], [(1,15),(2,11),(3,17),(4,13),(5,19),(6,14),(7,20),(8,16),(9,12),(10,18),(21,36),(22,27),(23,38),(24,29),(25,40),(26,31),(28,33),(30,35),(32,37),(34,39)], [(1,7),(2,8),(3,9),(4,10),(5,6),(11,16),(12,17),(13,18),(14,19),(15,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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)], [(1,2),(3,5),(6,9),(7,8),(11,15),(12,14),(16,20),(17,19),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40)]])
38 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | 10B | 10C | ··· | 10H | 10I | 10J | 20A | ··· | 20J |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
size | 1 | 1 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 4 | 8 | 8 | 20 | 20 | 40 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
38 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D20 | D20 | C2wrC22 | D4xD5 | C23:D20 |
kernel | C23:D20 | C23.1D10 | C5xC23:C4 | C22:D20 | C23:D10 | D4:6D10 | C2xDic5 | C2xC20 | C22xD5 | C22xC10 | C23:C4 | C22:C4 | C2xD4 | C2xC4 | C23 | C5 | C22 | C1 |
# reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 2 |
Matrix representation of C23:D20 ►in GL8(F41)
1 | 0 | 38 | 23 | 0 | 0 | 0 | 0 |
0 | 1 | 36 | 23 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
40 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
40 | 33 | 1 | 40 | 0 | 0 | 0 | 0 |
7 | 6 | 36 | 6 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 40 | 1 | 0 | 0 | 0 | 0 | 0 |
34 | 39 | 36 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,38,36,40,0,0,0,0,0,23,23,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,40,40,7,0,0,0,0,1,34,33,6,0,0,0,0,0,0,1,36,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,1,1,34,0,0,0,0,1,0,40,39,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C23:D20 in GAP, Magma, Sage, TeX
C_2^3\rtimes D_{20}
% in TeX
G:=Group("C2^3:D20");
// GroupNames label
G:=SmallGroup(320,368);
// by ID
G=gap.SmallGroup(320,368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,570,1684,438,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations