Copied to
clipboard

G = C24⋊2Dic5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — C24⋊2Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — D4×C10 — C23⋊Dic5 — C24⋊2Dic5
 Lower central C5 — C10 — C2×C10 — C22×C10 — C24⋊2Dic5
 Upper central C1 — C2 — C22 — C2×D4 — C22≀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, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C22⋊C4, M4(2), C2×D4, C2×D4, C24, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C4.D4, C22≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, 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, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C23⋊C4, C2×Dic5, C5⋊D4, C2≀C4, C23.D5, C23⋊Dic5, C242Dic5

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

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 5A 5B 8A 8B 10A ··· 10F 10G ··· 10R 10S 10T 20A ··· 20F order 1 2 2 2 2 2 2 4 4 4 4 5 5 8 8 10 ··· 10 10 ··· 10 10 10 20 ··· 20 size 1 1 2 4 4 4 4 4 8 40 40 2 2 40 40 2 ··· 2 4 ··· 4 8 8 8 ··· 8

41 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + - + + image C1 C2 C2 C2 C4 C4 D4 D4 D5 Dic5 D10 Dic5 C5⋊D4 C5⋊D4 C23⋊C4 C2≀C4 C23⋊Dic5 C24⋊2Dic5 kernel C24⋊2Dic5 C20.D4 C23⋊Dic5 C5×C22≀C2 C5×C22⋊C4 C23×C10 C2×C20 C22×C10 C22≀C2 C22⋊C4 C2×D4 C24 C2×C4 C23 C10 C5 C2 C1 # reps 1 1 1 1 2 2 1 1 2 2 2 2 4 4 1 2 4 8

Matrix representation of C242Dic5 in GL6(𝔽41)

 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 39 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 39 0 0 0 0 0 1 0 0 0 0 0 0 40 39 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 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 40 0 0 0 0 25 16 0 0 0 0 0 0 1 0 0 0 0 0 40 40 0 0 0 0 0 0 1 0 0 0 0 0 40 40
,
 7 32 0 0 0 0 1 34 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 40 40 0 0

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