Copied to
clipboard

G = C242Dic5order 320 = 26·5

1st semidirect product of C24 and Dic5 acting via Dic5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C242Dic5, C55C2≀C4, (C2×C20).3D4, (C23×C10)⋊3C4, (C2×D4).7D10, C20.D42C2, C22≀C2.1D5, C22⋊C41Dic5, C23⋊Dic52C2, (D4×C10).5C22, (C22×C10).14D4, C23.5(C5⋊D4), C23.1(C2×Dic5), C10.40(C23⋊C4), C2.4(C23⋊Dic5), C22.12(C23.D5), (C5×C22⋊C4)⋊7C4, (C2×C4).5(C5⋊D4), (C5×C22≀C2).1C2, (C22×C10).38(C2×C4), (C2×C10).158(C22⋊C4), SmallGroup(320,94)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C242Dic5
C1C5C10C2×C10C22×C10D4×C10C23⋊Dic5 — C242Dic5
C5C10C2×C10C22×C10 — C242Dic5
C1C2C22C2×D4C22≀C2

Generators and relations for C242Dic5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=e5, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 350 in 94 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2 [×5], C4 [×3], C22, C22 [×12], C5, C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], C10, C10 [×5], C22⋊C4, C22⋊C4 [×2], M4(2), C2×D4, C2×D4, C24, Dic5, C20 [×2], C2×C10, C2×C10 [×12], C23⋊C4, C4.D4, C22≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4 [×3], C22×C10 [×2], C22×C10 [×4], C2≀C4, C4.Dic5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, D4×C10, D4×C10, C23×C10, C20.D4, C23⋊Dic5, C5×C22≀C2, C242Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C2≀C4, C23.D5, C23⋊Dic5, C242Dic5

Smallest permutation representation of C242Dic5
On 40 points
Generators in S40
(11 25)(12 21)(13 27)(14 23)(15 29)(16 26)(17 22)(18 28)(19 24)(20 30)
(1 31)(2 37)(3 33)(4 39)(5 35)(6 38)(7 34)(8 40)(9 36)(10 32)(11 25)(12 21)(13 27)(14 23)(15 29)(16 26)(17 22)(18 28)(19 24)(20 30)
(11 20)(12 16)(13 17)(14 18)(15 19)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 20)(12 16)(13 17)(14 18)(15 19)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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 17)(2 16)(3 20)(4 19)(5 18)(6 11)(7 15)(8 14)(9 13)(10 12)(21 32 26 37)(22 31 27 36)(23 40 28 35)(24 39 29 34)(25 38 30 33)

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

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

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

41 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D5A5B8A8B10A···10F10G···10R10S10T20A···20F
order12222224444558810···1010···10101020···20
size11244444840402240402···24···4888···8

41 irreducible representations

dim111111222222224444
type+++++++-+-++
imageC1C2C2C2C4C4D4D4D5Dic5D10Dic5C5⋊D4C5⋊D4C23⋊C4C2≀C4C23⋊Dic5C242Dic5
kernelC242Dic5C20.D4C23⋊Dic5C5×C22≀C2C5×C22⋊C4C23×C10C2×C20C22×C10C22≀C2C22⋊C4C2×D4C24C2×C4C23C10C5C2C1
# reps111122112222441248

Matrix representation of C242Dic5 in GL6(𝔽41)

100000
010000
001000
000100
00004039
000001
,
100000
010000
00403900
000100
00004039
000001
,
100000
010000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
19400000
25160000
001000
00404000
000010
00004040
,
7320000
1340000
000010
000001
001000
00404000

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[19,25,0,0,0,0,40,16,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[7,1,0,0,0,0,32,34,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C242Dic5 in GAP, Magma, Sage, TeX

C_2^4\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("C2^4:2Dic5");
// GroupNames label

G:=SmallGroup(320,94);
// by ID

G=gap.SmallGroup(320,94);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,675,297,1684,12550]);
// Polycyclic

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

׿
×
𝔽