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## G = C5×2+ 1+4order 160 = 25·5

### Direct product of C5 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×2+ 1+4
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — D4×C10 — C5×2+ 1+4
 Lower central C1 — C2 — C5×2+ 1+4
 Upper central C1 — C10 — C5×2+ 1+4

Generators and relations for C5×2+ 1+4
G = < a,b,c,d,e | a5=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C10, C10, C2×D4, C4○D4, C20, C2×C10, C2×C10, 2+ 1+4, C2×C20, C5×D4, C5×Q8, C22×C10, D4×C10, C5×C4○D4, C5×2+ 1+4
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, 2+ 1+4, C22×C10, C23×C10, C5×2+ 1+4

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

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

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

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

C5×2+ 1+4 is a maximal subgroup of   2+ 1+4⋊D5  2+ 1+4.D5  2+ 1+4.2D5  2+ 1+42D5  D20.32C23  D20.33C23  D20.37C23
C5×2+ 1+4 is a maximal quotient of   C5×D42  C5×Q82

85 conjugacy classes

 class 1 2A 2B ··· 2J 4A ··· 4F 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10AN 20A ··· 20X order 1 2 2 ··· 2 4 ··· 4 5 5 5 5 10 10 10 10 10 ··· 10 20 ··· 20 size 1 1 2 ··· 2 2 ··· 2 1 1 1 1 1 1 1 1 2 ··· 2 2 ··· 2

85 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C5 C10 C10 2+ 1+4 C5×2+ 1+4 kernel C5×2+ 1+4 D4×C10 C5×C4○D4 2+ 1+4 C2×D4 C4○D4 C5 C1 # reps 1 9 6 4 36 24 1 4

Matrix representation of C5×2+ 1+4 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 0 0 1 0 40 1 40 2 40 0 0 0 0 40 1 40
,
 0 1 0 0 1 0 0 0 1 40 1 39 0 0 0 40
,
 0 0 40 0 40 1 40 2 1 0 0 0 1 40 0 40
,
 0 1 0 0 1 0 0 0 40 1 40 2 40 1 0 1
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[0,40,40,0,0,1,0,40,1,40,0,1,0,2,0,40],[0,1,1,0,1,0,40,0,0,0,1,0,0,0,39,40],[0,40,1,1,0,1,0,40,40,40,0,0,0,2,0,40],[0,1,40,40,1,0,1,1,0,0,40,0,0,0,2,1] >;

C5×2+ 1+4 in GAP, Magma, Sage, TeX

C_5\times 2_+^{1+4}
% in TeX

G:=Group("C5xES+(2,2)");
// GroupNames label

G:=SmallGroup(160,232);
// by ID

G=gap.SmallGroup(160,232);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985,764,2115]);
// Polycyclic

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

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