Copied to
clipboard

G = C2xDic5:2D5order 400 = 24·52

Direct product of C2 and Dic5:2D5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2xDic5:2D5, Dic5:6D10, C102.9C22, C10:2(C4xD5), C22.9D52, (C2xDic5):5D5, C52:9(C22xC4), (C10xDic5):8C2, (C2xC10).13D10, (C5xC10).13C23, (C5xDic5):7C22, C10.13(C22xD5), C5:3(C2xC4xD5), C2.3(C2xD52), C5:D5:4(C2xC4), (C2xC5:D5):5C4, (C5xC10):8(C2xC4), (C22xC5:D5).3C2, (C2xC5:D5).17C22, SmallGroup(400,175)

Series: Derived Chief Lower central Upper central

C1C52 — C2xDic5:2D5
C1C5C52C5xC10C5xDic5Dic5:2D5 — C2xDic5:2D5
C52 — C2xDic5:2D5
C1C22

Generators and relations for C2xDic5:2D5
 G = < a,b,c,d,e | a2=b10=d5=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 908 in 140 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2xC4, C23, D5, C10, C10, C22xC4, Dic5, C20, D10, C2xC10, C2xC10, C52, C4xD5, C2xDic5, C2xC20, C22xD5, C5:D5, C5xC10, C5xC10, C2xC4xD5, C5xDic5, C2xC5:D5, C102, Dic5:2D5, C10xDic5, C22xC5:D5, C2xDic5:2D5
Quotients: C1, C2, C4, C22, C2xC4, C23, D5, C22xC4, D10, C4xD5, C22xD5, C2xC4xD5, D52, Dic5:2D5, C2xD52, C2xDic5:2D5

Smallest permutation representation of C2xDic5:2D5
On 40 points
Generators in S40
(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 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 24 6 29)(2 23 7 28)(3 22 8 27)(4 21 9 26)(5 30 10 25)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)
(1 5 9 3 7)(2 6 10 4 8)(11 15 19 13 17)(12 16 20 14 18)(21 27 23 29 25)(22 28 24 30 26)(31 37 33 39 35)(32 38 34 40 36)
(1 7)(2 6)(3 5)(8 10)(11 17)(12 16)(13 15)(18 20)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)

G:=sub<Sym(40)| (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), (1,2,3,4,5,6,7,8,9,10)(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), (1,24,6,29)(2,23,7,28)(3,22,8,27)(4,21,9,26)(5,30,10,25)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35), (1,5,9,3,7)(2,6,10,4,8)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,37,33,39,35)(32,38,34,40,36), (1,7)(2,6)(3,5)(8,10)(11,17)(12,16)(13,15)(18,20)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)>;

G:=Group( (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), (1,2,3,4,5,6,7,8,9,10)(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), (1,24,6,29)(2,23,7,28)(3,22,8,27)(4,21,9,26)(5,30,10,25)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35), (1,5,9,3,7)(2,6,10,4,8)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,37,33,39,35)(32,38,34,40,36), (1,7)(2,6)(3,5)(8,10)(11,17)(12,16)(13,15)(18,20)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37) );

G=PermutationGroup([[(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)], [(1,2,3,4,5,6,7,8,9,10),(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)], [(1,24,6,29),(2,23,7,28),(3,22,8,27),(4,21,9,26),(5,30,10,25),(11,34,16,39),(12,33,17,38),(13,32,18,37),(14,31,19,36),(15,40,20,35)], [(1,5,9,3,7),(2,6,10,4,8),(11,15,19,13,17),(12,16,20,14,18),(21,27,23,29,25),(22,28,24,30,26),(31,37,33,39,35),(32,38,34,40,36)], [(1,7),(2,6),(3,5),(8,10),(11,17),(12,16),(13,15),(18,20),(22,30),(23,29),(24,28),(25,27),(32,40),(33,39),(34,38),(35,37)]])

64 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H5A5B5C5D5E5F5G5H10A···10L10M···10X20A···20P
order122222224···45555555510···1010···1020···20
size1111252525255···5222244442···24···410···10

64 irreducible representations

dim111112222444
type++++++++++
imageC1C2C2C2C4D5D10D10C4xD5D52Dic5:2D5C2xD52
kernelC2xDic5:2D5Dic5:2D5C10xDic5C22xC5:D5C2xC5:D5C2xDic5Dic5C2xC10C10C22C2C2
# reps1421848416484

Matrix representation of C2xDic5:2D5 in GL6(F41)

100000
010000
0040000
0004000
000010
000001
,
710000
4000000
0064000
001000
000010
000001
,
900000
19320000
001000
0064000
000010
000001
,
100000
010000
001000
000100
0000640
000010
,
100000
34400000
001000
0064000
0000640
00003535

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,40,0,0,0,0,1,0,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,19,0,0,0,0,0,32,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0],[1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,40,35] >;

C2xDic5:2D5 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5\rtimes_2D_5
% in TeX

G:=Group("C2xDic5:2D5");
// GroupNames label

G:=SmallGroup(400,175);
// by ID

G=gap.SmallGroup(400,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,55,970,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=d^5=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<