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G = C2×Dic52D5order 400 = 24·52

Direct product of C2 and Dic52D5

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

Aliases: C2×Dic52D5, Dic56D10, C102.9C22, C102(C4×D5), C22.9D52, (C2×Dic5)⋊5D5, C529(C22×C4), (C10×Dic5)⋊8C2, (C2×C10).13D10, (C5×C10).13C23, (C5×Dic5)⋊7C22, C10.13(C22×D5), C53(C2×C4×D5), C2.3(C2×D52), C5⋊D54(C2×C4), (C2×C5⋊D5)⋊5C4, (C5×C10)⋊8(C2×C4), (C22×C5⋊D5).3C2, (C2×C5⋊D5).17C22, SmallGroup(400,175)

Series: Derived Chief Lower central Upper central

C1C52 — C2×Dic52D5
C1C5C52C5×C10C5×Dic5Dic52D5 — C2×Dic52D5
C52 — C2×Dic52D5
C1C22

Generators and relations for C2×Dic52D5
 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, C2×C4, C23, D5, C10, C10, C22×C4, Dic5, C20, D10, C2×C10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C22×D5, C5⋊D5, C5×C10, C5×C10, C2×C4×D5, C5×Dic5, C2×C5⋊D5, C102, Dic52D5, C10×Dic5, C22×C5⋊D5, C2×Dic52D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C4×D5, C22×D5, C2×C4×D5, D52, Dic52D5, C2×D52, C2×Dic52D5

Smallest permutation representation of C2×Dic52D5
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++++++++++
imageC1C2C2C2C4D5D10D10C4×D5D52Dic52D5C2×D52
kernelC2×Dic52D5Dic52D5C10×Dic5C22×C5⋊D5C2×C5⋊D5C2×Dic5Dic5C2×C10C10C22C2C2
# reps1421848416484

Matrix representation of C2×Dic52D5 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] >;

C2×Dic52D5 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

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