Copied to
clipboard

G = C10.D20  order 400 = 24·52

6th non-split extension by C10 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial

Aliases: C10.17D20, C102.3C22, C22.5D52, (C2×Dic5)⋊2D5, (C5×C10).14D4, C10.10(C4×D5), (C2×C10).9D10, (C10×Dic5)⋊2C2, C10.5(C5⋊D4), C52⋊9(C22⋊C4), C5⋊2(D10⋊C4), C2.2(C5⋊D20), C2.4(Dic5⋊2D5), (C2×C5⋊D5)⋊3C4, (C5×C10).47(C2×C4), (C22×C5⋊D5).1C2, SmallGroup(400,73)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — C10.D20
Chief series
C1 — C5 — C52 — C5×C10 — C102 — C10×Dic5 — C10.D20
Lower central
C52 — C5×C10 — C10.D20
Upper central
C1 — C22

Generators and relations for C10.D20
 G = < a,b,c | a10=b20=c2=1, bab-1=cac=a-1, cbc=a5b-1 >

Subgroups: 764 in 100 conjugacy classes, 28 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, C10, C10, C22⋊C4, Dic5, C20, D10, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C22×D5, C5⋊D5, C5×C10, C5×C10, D10⋊C4, C5×Dic5, C2×C5⋊D5, C2×C5⋊D5, C102, C10×Dic5, C22×C5⋊D5, C10.D20
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C4×D5, D20, C5⋊D4, D10⋊C4, D52, Dic5⋊2D5, C5⋊D20, C10.D20

Smallest permutation representation of C10.D20
►On 40 points
Generators in S40
(1 30 13 22 5 34 17 26 9 38)(2 39 10 27 18 35 6 23 14 31)(3 32 15 24 7 36 19 28 11 40)(4 21 12 29 20 37 8 25 16 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 15)(2 27)(3 13)(4 25)(5 11)(6 23)(7 9)(8 21)(10 39)(12 37)(14 35)(16 33)(17 19)(18 31)(20 29)(22 40)(24 38)(26 36)(28 34)(30 32)
 
G:=sub<Sym(40)| (1,30,13,22,5,34,17,26,9,38)(2,39,10,27,18,35,6,23,14,31)(3,32,15,24,7,36,19,28,11,40)(4,21,12,29,20,37,8,25,16,33), (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), (1,15)(2,27)(3,13)(4,25)(5,11)(6,23)(7,9)(8,21)(10,39)(12,37)(14,35)(16,33)(17,19)(18,31)(20,29)(22,40)(24,38)(26,36)(28,34)(30,32)>;
 
G:=Group( (1,30,13,22,5,34,17,26,9,38)(2,39,10,27,18,35,6,23,14,31)(3,32,15,24,7,36,19,28,11,40)(4,21,12,29,20,37,8,25,16,33), (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), (1,15)(2,27)(3,13)(4,25)(5,11)(6,23)(7,9)(8,21)(10,39)(12,37)(14,35)(16,33)(17,19)(18,31)(20,29)(22,40)(24,38)(26,36)(28,34)(30,32) );
 
G=PermutationGroup([[(1,30,13,22,5,34,17,26,9,38),(2,39,10,27,18,35,6,23,14,31),(3,32,15,24,7,36,19,28,11,40),(4,21,12,29,20,37,8,25,16,33)], [(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)], [(1,15),(2,27),(3,13),(4,25),(5,11),(6,23),(7,9),(8,21),(10,39),(12,37),(14,35),(16,33),(17,19),(18,31),(20,29),(22,40),(24,38),(26,36),(28,34),(30,32)]])
 

58 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B5C5D5E5F5G5H10A···10L10M···10X20A···20P
order12222244445555555510···1010···1020···20
size1111505010101010222244442···24···410···10

58 irreducible representations

dim1111222222444
type++++++++++
imageC1C2C2C4D4D5D10C4×D5D20C5⋊D4D52Dic5⋊2D5C5⋊D20
kernelC10.D20C10×Dic5C22×C5⋊D5C2×C5⋊D5C5×C10C2×Dic5C2×C10C10C10C10C22C2C2
# reps1214244888448

Matrix representation of C10.D20 ►in GL6(𝔽41)

4000000
0400000
001000
000100
0000406
00003535
,
1390000
0400000
00253900
0021300
000010
0000640
,
100000
1400000
0035100
006600
000010
0000640

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,6,35],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,25,2,0,0,0,0,39,13,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
 

C10.D20 in GAP, Magma, Sage, TeX 

C_{10}.D_{20}
 
% in TeX
 
G:=Group("C10.D20");
 
// GroupNames label
 
G:=SmallGroup(400,73);
 
// by ID
 
G=gap.SmallGroup(400,73);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,79,970,11525]);
 
// Polycyclic
 
G:=Group<a,b,c|a^10=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^5*b^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁