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## G = C5×C2≀C22order 320 = 26·5

### Direct product of C5 and C2≀C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C5×C2≀C22
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C23×C10 — C5×C22≀C2 — C5×C2≀C22
 Lower central C1 — C2 — C23 — C5×C2≀C22
 Upper central C1 — C10 — C22×C10 — C5×C2≀C22

Generators and relations for C5×C2≀C22
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 450 in 198 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, C20, C2×C10, C2×C10, C23⋊C4, C22≀C2, 2+ 1+4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C2≀C22, C5×C22⋊C4, C5×C22⋊C4, D4×C10, D4×C10, C5×C4○D4, C23×C10, C5×C23⋊C4, C5×C22≀C2, C5×2+ 1+4, C5×C2≀C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C5×D4, C22×C10, C2≀C22, D4×C10, C5×C22≀C2, C5×C2≀C22

Smallest permutation representation of C5×C2≀C22
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 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 31)(12 32)(13 33)(14 34)(15 35)(16 26)(17 27)(18 28)(19 29)(20 30)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(21 31)(22 32)(23 33)(24 34)(25 35)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 36 26 11)(2 37 27 12)(3 38 28 13)(4 39 29 14)(5 40 30 15)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)
(6 16)(7 17)(8 18)(9 19)(10 20)(11 36)(12 37)(13 38)(14 39)(15 40)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2I 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10AJ 20A ··· 20L 20M ··· 20X order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 5 5 5 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 2 2 4 ··· 4 4 4 4 8 8 8 1 1 1 1 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D4 C5×D4 C5×D4 C2≀C22 C5×C2≀C22 kernel C5×C2≀C22 C5×C23⋊C4 C5×C22≀C2 C5×2+ 1+4 C2≀C22 C23⋊C4 C22≀C2 2+ 1+4 C2×C20 C22×C10 C2×C4 C23 C5 C1 # reps 1 3 3 1 4 12 12 4 3 3 12 12 2 8

Matrix representation of C5×C2≀C22 in GL6(𝔽41)

 10 0 0 0 0 0 0 10 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 1 0 0 0 0 0 0 40 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0
,
 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
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0
,
 40 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 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C5×C2≀C22 in GAP, Magma, Sage, TeX

C_5\times C_2\wr C_2^2
% in TeX

G:=Group("C5xC2wrC2^2");
// GroupNames label

G:=SmallGroup(320,958);
// by ID

G=gap.SmallGroup(320,958);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,1768,5052]);
// Polycyclic

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

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