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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

Aliases: C5×C2≀C22, 2+ 1+42C10, C23⋊(C5×D4), (C2×C20)⋊4D4, C23⋊C43C10, C242(C2×C10), (C22×C10)⋊1D4, C22≀C21C10, (C23×C10)⋊1C22, C22.18(D4×C10), C10.104C22≀C2, C23.3(C22×C10), (C5×2+ 1+4)⋊8C2, (D4×C10).183C22, (C22×C10).82C23, (C2×C4)⋊(C5×D4), (C5×C23⋊C4)⋊9C2, C22⋊C41(C2×C10), (C2×D4).8(C2×C10), (C5×C22≀C2)⋊11C2, C2.18(C5×C22≀C2), (C2×C10).413(C2×D4), (C5×C22⋊C4)⋊36C22, SmallGroup(320,958)

Series: Derived Chief Lower central Upper central

C1C23 — C5×C2≀C22
C1C2C22C23C22×C10C23×C10C5×C22≀C2 — C5×C2≀C22
C1C2C23 — C5×C2≀C22
C1C10C22×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 [×8], C4 [×6], C22 [×3], C22 [×18], C5, C2×C4 [×3], C2×C4 [×6], D4 [×15], Q8, C23, C23 [×3], C23 [×6], C10, C10 [×8], C22⋊C4 [×3], C22⋊C4 [×3], C2×D4 [×3], C2×D4 [×6], C4○D4 [×3], C24, C20 [×6], C2×C10 [×3], C2×C10 [×18], C23⋊C4 [×3], C22≀C2 [×3], 2+ 1+4, C2×C20 [×3], C2×C20 [×6], C5×D4 [×15], C5×Q8, C22×C10, C22×C10 [×3], C22×C10 [×6], C2≀C22, C5×C22⋊C4 [×3], C5×C22⋊C4 [×3], D4×C10 [×3], D4×C10 [×6], C5×C4○D4 [×3], C23×C10, C5×C23⋊C4 [×3], C5×C22≀C2 [×3], C5×2+ 1+4, C5×C2≀C22
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], C23, C10 [×7], C2×D4 [×3], C2×C10 [×7], C22≀C2, C5×D4 [×6], C22×C10, C2≀C22, D4×C10 [×3], 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 26)(7 27)(8 28)(9 29)(10 30)(11 22)(12 23)(13 24)(14 25)(15 21)(16 35)(17 31)(18 32)(19 33)(20 34)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 14)(7 15)(8 11)(9 12)(10 13)(16 36)(17 37)(18 38)(19 39)(20 40)(21 27)(22 28)(23 29)(24 30)(25 26)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 37)(12 38)(13 39)(14 40)(15 36)(26 34)(27 35)(28 31)(29 32)(30 33)
(1 36 35 7)(2 37 31 8)(3 38 32 9)(4 39 33 10)(5 40 34 6)(11 28 17 22)(12 29 18 23)(13 30 19 24)(14 26 20 25)(15 27 16 21)
(6 40)(7 36)(8 37)(9 38)(10 39)(11 17)(12 18)(13 19)(14 20)(15 16)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E···2I4A4B4C4D4E4F5A5B5C5D10A10B10C10D10E···10P10Q···10AJ20A···20L20M···20X
order122222···244444455551010101010···1010···1020···2020···20
size112224···4444888111111112···24···44···48···8

80 irreducible representations

dim11111111222244
type+++++++
imageC1C2C2C2C5C10C10C10D4D4C5×D4C5×D4C2≀C22C5×C2≀C22
kernelC5×C2≀C22C5×C23⋊C4C5×C22≀C2C5×2+ 1+4C2≀C22C23⋊C4C22≀C22+ 1+4C2×C20C22×C10C2×C4C23C5C1
# reps133141212433121228

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

1000000
0100000
001000
000100
000010
000001
,
100000
010000
000010
0000040
001000
0004000
,
100000
010000
000100
001000
0000040
0000400
,
100000
010000
0040000
0004000
0000400
0000040
,
010000
4000000
000001
000010
001000
000100
,
4000000
010000
001000
000100
000001
000010

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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