metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊D20, (C2×C4)⋊D20, C5⋊1C2≀C22, (C2×C20)⋊1D4, C23⋊C4⋊3D5, C22⋊C4⋊1D10, (C2×Dic5)⋊1D4, (C22×D5)⋊1D4, (C22×C10)⋊2D4, C23⋊D10⋊1C2, D4⋊6D10⋊1C2, C22⋊D20⋊1C2, (C2×D4).11D10, C22.8(C2×D20), C22.24(D4×D5), C10.13C22≀C2, (D4×C10).8C22, (C23×D5)⋊1C22, C23.D5⋊1C22, C23.2(C22×D5), C23.1D10⋊1C2, (C22×C10).2C23, C2.16(C22⋊D20), (C5×C23⋊C4)⋊4C2, (C2×C10).17(C2×D4), (C5×C22⋊C4)⋊1C22, (C2×C5⋊D4).2C22, SmallGroup(320,368)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C5 — C10 — C2×C10 — C22×C10 — C2×C5⋊D4 — D4⋊6D10 — C23⋊D20 |
C1 — C2 — C23 — C23⋊C4 |
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 [×8], C4 [×6], C22, C22 [×2], C22 [×18], C5, C2×C4, C2×C4 [×8], D4 [×15], Q8, C23 [×2], C23 [×8], D5 [×4], C10, C10 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C2×D4, C2×D4 [×8], C4○D4 [×3], C24, Dic5 [×3], C20 [×3], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×3], C23⋊C4, C23⋊C4 [×2], C22≀C2 [×3], 2+ 1+4, Dic10, C4×D5 [×2], D20 [×5], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10 [×2], C2≀C22, D10⋊C4 [×3], C23.D5, C5×C22⋊C4 [×2], C2×D20 [×2], C4○D20, D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×2], C2×C5⋊D4 [×2], D4×C10, C23×D5, C23.1D10 [×2], C5×C23⋊C4, C22⋊D20 [×2], C23⋊D10, D4⋊6D10, C23⋊D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, C2≀C22, C2×D20, D4×D5 [×2], C22⋊D20, C23⋊D20
(1 24)(2 40)(3 36)(4 32)(5 28)(6 38)(7 34)(8 30)(9 26)(10 22)(11 29)(12 35)(13 21)(14 27)(15 33)(16 39)(17 25)(18 31)(19 37)(20 23)
(1 11)(2 17)(3 13)(4 19)(5 15)(6 20)(7 16)(8 12)(9 18)(10 14)(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 5)(2 4)(6 7)(8 10)(11 15)(12 14)(16 20)(17 19)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)
G:=sub<Sym(40)| (1,24)(2,40)(3,36)(4,32)(5,28)(6,38)(7,34)(8,30)(9,26)(10,22)(11,29)(12,35)(13,21)(14,27)(15,33)(16,39)(17,25)(18,31)(19,37)(20,23), (1,11)(2,17)(3,13)(4,19)(5,15)(6,20)(7,16)(8,12)(9,18)(10,14)(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), (1,5)(2,4)(6,7)(8,10)(11,15)(12,14)(16,20)(17,19)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)>;
G:=Group( (1,24)(2,40)(3,36)(4,32)(5,28)(6,38)(7,34)(8,30)(9,26)(10,22)(11,29)(12,35)(13,21)(14,27)(15,33)(16,39)(17,25)(18,31)(19,37)(20,23), (1,11)(2,17)(3,13)(4,19)(5,15)(6,20)(7,16)(8,12)(9,18)(10,14)(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), (1,5)(2,4)(6,7)(8,10)(11,15)(12,14)(16,20)(17,19)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34) );
G=PermutationGroup([(1,24),(2,40),(3,36),(4,32),(5,28),(6,38),(7,34),(8,30),(9,26),(10,22),(11,29),(12,35),(13,21),(14,27),(15,33),(16,39),(17,25),(18,31),(19,37),(20,23)], [(1,11),(2,17),(3,13),(4,19),(5,15),(6,20),(7,16),(8,12),(9,18),(10,14),(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)], [(1,5),(2,4),(6,7),(8,10),(11,15),(12,14),(16,20),(17,19),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34)])
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 | C2≀C22 | D4×D5 | C23⋊D20 |
kernel | C23⋊D20 | C23.1D10 | C5×C23⋊C4 | C22⋊D20 | C23⋊D10 | D4⋊6D10 | C2×Dic5 | C2×C20 | C22×D5 | C22×C10 | C23⋊C4 | C22⋊C4 | C2×D4 | C2×C4 | 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