Copied to
clipboard

## G = C5×C4≀C2order 160 = 25·5

### Direct product of C5 and C4≀C2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C4≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4≀C2
 Lower central C1 — C2 — C4 — C5×C4≀C2
 Upper central C1 — C20 — C2×C20 — C5×C4≀C2

Generators and relations for C5×C4≀C2
G = < a,b,c,d | a5=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Smallest permutation representation of C5×C4≀C2
On 40 points
Generators in S40
(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 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 16 36 11)(7 17 37 12)(8 18 38 13)(9 19 39 14)(10 20 40 15)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 16 36 11)(7 17 37 12)(8 18 38 13)(9 19 39 14)(10 20 40 15)(26 31)(27 32)(28 33)(29 34)(30 35)

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

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

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

C5×C4≀C2 is a maximal subgroup of   C42⋊D10  D44D20  M4(2).22D10  C42.196D10  M4(2)⋊D10  D4.9D20  D4.10D20

70 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4G 4H 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A ··· 20H 20I ··· 20AB 20AC 20AD 20AE 20AF 40A ··· 40H order 1 2 2 2 4 4 4 ··· 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 4 1 1 2 ··· 2 4 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 4 4 4 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 D4 C4≀C2 C5×D4 C5×D4 C5×C4≀C2 kernel C5×C4≀C2 C4×C20 C5×M4(2) C5×C4○D4 C5×D4 C5×Q8 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C20 C2×C10 C5 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 4 4 4 16

Matrix representation of C5×C4≀C2 in GL2(𝔽41) generated by

 37 0 0 37
,
 9 0 0 32
,
 0 32 9 0
,
 40 0 0 32
G:=sub<GL(2,GF(41))| [37,0,0,37],[9,0,0,32],[0,9,32,0],[40,0,0,32] >;

C5×C4≀C2 in GAP, Magma, Sage, TeX

C_5\times C_4\wr C_2
% in TeX

G:=Group("C5xC4wrC2");
// GroupNames label

G:=SmallGroup(160,54);
// by ID

G=gap.SmallGroup(160,54);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,2403,1209,117,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations

Export

׿
×
𝔽