C5xC2^4:C5 - GroupNames

G = C₅×C₂⁴⋊C₅ order 400 = 2⁴·5²

Direct product of C₅ and C₂⁴⋊C₅

direct product, metabelian, soluble, monomial, A-group

Aliases: C₅×C₂⁴⋊C₅, C₂⁴⋊C₅², (C₂³×C₁₀)⋊C₅, SmallGroup(400,213)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₅×C₂⁴⋊C₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₅×C₂⁴⋊C₅

Lower central

C₂⁴ — C₅×C₂⁴⋊C₅

Upper central

C₁ — C₅

Generators and relations for C₅×C₂⁴⋊C₅
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e²=f⁵=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bcd, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, fef^-1=b >

C₁

₅C₂

C₅

₁₆C₅

₅C₂²

₅C₁₀

₁₆C₅²

₅C₂³

₅C₂×C₁₀

C₂⁴

₅C₂²×C₁₀

C₂⁴⋊C₅

C₂³×C₁₀

C₂⁴⋊C₅

C₅×C₂⁴⋊C₅

Smallest permutation representation of C₅×C₂⁴⋊C₅
►On 50 points

Generators in S₅₀

(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)(41 42 43 44 45)(46 47 48 49 50)

(1 13)(2 14)(3 15)(4 11)(5 12)(36 42)(37 43)(38 44)(39 45)(40 41)

(1 13)(2 14)(3 15)(4 11)(5 12)(6 50)(7 46)(8 47)(9 48)(10 49)(16 22)(17 23)(18 24)(19 25)(20 21)(26 32)(27 33)(28 34)(29 35)(30 31)

(26 32)(27 33)(28 34)(29 35)(30 31)(36 42)(37 43)(38 44)(39 45)(40 41)

(6 50)(7 46)(8 47)(9 48)(10 49)(26 32)(27 33)(28 34)(29 35)(30 31)

(1 46 40 29 18)(2 47 36 30 19)(3 48 37 26 20)(4 49 38 27 16)(5 50 39 28 17)(6 45 34 23 12)(7 41 35 24 13)(8 42 31 25 14)(9 43 32 21 15)(10 44 33 22 11)

G:=sub<Sym(50)| (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)(41,42,43,44,45)(46,47,48,49,50), (1,13)(2,14)(3,15)(4,11)(5,12)(36,42)(37,43)(38,44)(39,45)(40,41), (1,13)(2,14)(3,15)(4,11)(5,12)(6,50)(7,46)(8,47)(9,48)(10,49)(16,22)(17,23)(18,24)(19,25)(20,21)(26,32)(27,33)(28,34)(29,35)(30,31), (26,32)(27,33)(28,34)(29,35)(30,31)(36,42)(37,43)(38,44)(39,45)(40,41), (6,50)(7,46)(8,47)(9,48)(10,49)(26,32)(27,33)(28,34)(29,35)(30,31), (1,46,40,29,18)(2,47,36,30,19)(3,48,37,26,20)(4,49,38,27,16)(5,50,39,28,17)(6,45,34,23,12)(7,41,35,24,13)(8,42,31,25,14)(9,43,32,21,15)(10,44,33,22,11)>;

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)(41,42,43,44,45)(46,47,48,49,50), (1,13)(2,14)(3,15)(4,11)(5,12)(36,42)(37,43)(38,44)(39,45)(40,41), (1,13)(2,14)(3,15)(4,11)(5,12)(6,50)(7,46)(8,47)(9,48)(10,49)(16,22)(17,23)(18,24)(19,25)(20,21)(26,32)(27,33)(28,34)(29,35)(30,31), (26,32)(27,33)(28,34)(29,35)(30,31)(36,42)(37,43)(38,44)(39,45)(40,41), (6,50)(7,46)(8,47)(9,48)(10,49)(26,32)(27,33)(28,34)(29,35)(30,31), (1,46,40,29,18)(2,47,36,30,19)(3,48,37,26,20)(4,49,38,27,16)(5,50,39,28,17)(6,45,34,23,12)(7,41,35,24,13)(8,42,31,25,14)(9,43,32,21,15)(10,44,33,22,11) );

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),(41,42,43,44,45),(46,47,48,49,50)], [(1,13),(2,14),(3,15),(4,11),(5,12),(36,42),(37,43),(38,44),(39,45),(40,41)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,50),(7,46),(8,47),(9,48),(10,49),(16,22),(17,23),(18,24),(19,25),(20,21),(26,32),(27,33),(28,34),(29,35),(30,31)], [(26,32),(27,33),(28,34),(29,35),(30,31),(36,42),(37,43),(38,44),(39,45),(40,41)], [(6,50),(7,46),(8,47),(9,48),(10,49),(26,32),(27,33),(28,34),(29,35),(30,31)], [(1,46,40,29,18),(2,47,36,30,19),(3,48,37,26,20),(4,49,38,27,16),(5,50,39,28,17),(6,45,34,23,12),(7,41,35,24,13),(8,42,31,25,14),(9,43,32,21,15),(10,44,33,22,11)]])

40 conjugacy classes

class	1	2A	2B	2C	5A	5B	5C	5D	5E	···	5X	10A	···	10L
order	1	2	2	2	5	5	5	5	5	···	5	10	···	10
size	1	5	5	5	1	1	1	1	16	···	16	5	···	5

40 irreducible representations

dim	1	1	1	5	5
type	+			+
image	C₁	C₅	C₅	C₂⁴⋊C₅	C₅×C₂⁴⋊C₅
kernel	C₅×C₂⁴⋊C₅	C₂⁴⋊C₅	C₂³×C₁₀	C₅	C₁
# reps	1	20	4	3	12

Matrix representation of C₅×C₂⁴⋊C₅ ►in GL₆(𝔽₁₁)

9	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	0
0	0	0	0	0	1

1	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	1	6	10	0
0	4	0	0	0	1

1	0	0	0	0	0
0	10	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	8	10	5	1	0
0	0	0	0	0	10

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	10	0	0
0	3	1	0	10	0
0	0	0	7	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	10	0	0
0	0	0	5	1	0
0	7	9	0	0	10

5	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	3	1	6	9	0
0	0	0	0	5	1
0	0	0	0	7	0

G:=sub<GL(6,GF(11))| [9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,10,0,0,0,4,0,0,1,0,1,0,0,0,0,1,6,0,0,0,0,0,10,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,10,0,0,8,0,0,0,10,0,10,0,0,0,0,10,5,0,0,0,0,0,1,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,3,0,0,0,1,0,1,0,0,0,0,10,0,7,0,0,0,0,10,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,7,0,0,1,0,0,9,0,0,0,10,5,0,0,0,0,0,1,0,0,0,0,0,0,10],[5,0,0,0,0,0,0,0,0,3,0,0,0,1,0,1,0,0,0,0,1,6,0,0,0,0,0,9,5,7,0,0,0,0,1,0] >;

C₅×C₂⁴⋊C₅ in GAP, Magma, Sage, TeX

C_5\times C_2^4\rtimes C_5

% in TeX

G:=Group("C5xC2^4:C5");

// GroupNames label

G:=SmallGroup(400,213);

// by ID

G=gap.SmallGroup(400,213);

# by ID

G:=PCGroup([6,-5,-5,-2,2,2,2,3602,5403,8254,13505]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₅×C₂⁴⋊C₅ in TeX

G = C5×C24⋊C5 order 400 = 24·52

Direct product of C5 and C24⋊C5

G = C₅×C₂⁴⋊C₅ order 400 = 2⁴·5²

Direct product of C₅ and C₂⁴⋊C₅