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G = C32⋊Dic10  order 360 = 23·32·5

The semidirect product of C32 and Dic10 acting via Dic10/C5=Q8

non-abelian, soluble, monomial

Aliases: C32⋊Dic10, C5⋊PSU3(𝔽2), (C3×C15)⋊2Q8, C3⋊S3.3D10, C32⋊C4.2D5, C32⋊Dic5.2C2, (C5×C32⋊C4).2C2, (C5×C3⋊S3).6C22, SmallGroup(360,136)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C5×C3⋊S3 — C32⋊Dic10
Chief series
C1 — C5 — C3×C15 — C5×C3⋊S3 — C32⋊Dic5 — C32⋊Dic10
Lower central
C3×C15 — C5×C3⋊S3 — C32⋊Dic10
Upper central
C1

Generators and relations for C32⋊Dic10
 G = < a,b,c,d | a3=b3=c20=1, d2=c10, ab=ba, cac-1=b, dad-1=ab-1, cbc-1=a-1, dbd-1=a-1b-1, dcd-1=c-1 >

C1
9C2
4C3
C5
9C4
45C4
45C4
12S3
C32
9C10
4C15
45Q8
C3⋊S3
9C20
9Dic5
9Dic5
12C5×S3
C3×C15
C32⋊C4
5C32⋊C4
5C32⋊C4
9Dic10
C5×C3⋊S3
5PSU3(𝔽2)
C32⋊Dic5
C32⋊Dic5
C5×C32⋊C4
C32⋊Dic10

Character table of C32⋊Dic10

 class 1234A4B4C5A5B10A10B15A15B15C15D20A20B20C20D
 size 198189090221818888818181818
ρ1111111111111111111    trivial
ρ21111-1-1111111111111    linear of order 2
ρ3111-1-1111111111-1-1-1-1    linear of order 2
ρ4111-11-111111111-1-1-1-1    linear of order 2
ρ5222-200-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ6222200-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ7222-200-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ8222200-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ92-2200022-2-222220000    symplectic lifted from Q8, Schur index 2
ρ102-22000-1-√5/2-1+√5/21+√5/21-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5    symplectic lifted from Dic10, Schur index 2
ρ112-22000-1+√5/2-1-√5/21-√5/21+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52    symplectic lifted from Dic10, Schur index 2
ρ122-22000-1-√5/2-1+√5/21+√5/21-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5    symplectic lifted from Dic10, Schur index 2
ρ132-22000-1+√5/2-1-√5/21-√5/21+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52    symplectic lifted from Dic10, Schur index 2
ρ1480-10008800-1-1-1-10000    orthogonal lifted from PSU3(𝔽2)
ρ1580-1000-2+2√5-2-2√500-2ζ53+ζ52-2ζ54+ζ5ζ54-2ζ5ζ53-2ζ520000    complex faithful
ρ1680-1000-2-2√5-2+2√500-2ζ54+ζ5ζ53-2ζ52-2ζ53+ζ52ζ54-2ζ50000    complex faithful
ρ1780-1000-2+2√5-2-2√500ζ53-2ζ52ζ54-2ζ5-2ζ54+ζ5-2ζ53+ζ520000    complex faithful
ρ1880-1000-2-2√5-2+2√500ζ54-2ζ5-2ζ53+ζ52ζ53-2ζ52-2ζ54+ζ50000    complex faithful

Smallest permutation representation of C32⋊Dic10
►On 45 points
Generators in S45
(1 42 32)(2 38 28)(3 34 44)(4 30 40)(5 26 36)(6 39 21)(7 45 12)(8 23 41)(9 14 27)(10 43 25)(11 29 16)(13 18 31)(15 33 20)(17 22 35)(19 37 24)

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

(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)

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

Matrix representation of C32⋊Dic10 ►in GL8(𝔽61)

1000026110
0100035260
000006001
00411041020
00101513760
2751600605900
11593030590
275151376060060
,
1026110000
0135260000
006001000
3129585958000
006000000
006000001
00410041120
00001523760
,
171000000
600000000
11593030600
3837520521440
00000010
00010000
0060170000
11594793206041
,
1744000000
6044000000
152692409241
26014510145158
00000001
00101523760
3032012047103
00100000

G:=sub<GL(8,GF(61))| [1,0,0,0,0,27,11,27,0,1,0,0,0,51,59,51,0,0,0,41,1,60,3,51,0,0,0,1,0,0,0,37,0,0,0,0,1,60,3,60,26,35,60,41,51,59,0,60,11,26,0,0,37,0,59,0,0,0,1,20,60,0,0,60],[1,0,0,31,0,0,0,0,0,1,0,29,0,0,0,0,26,35,60,58,60,60,41,0,11,26,0,59,0,0,0,0,0,0,1,58,0,0,0,1,0,0,0,0,0,0,41,52,0,0,0,0,0,0,1,37,0,0,0,0,0,1,20,60],[17,60,11,38,0,0,0,11,1,0,59,37,0,0,0,59,0,0,3,52,0,0,60,47,0,0,0,0,0,1,17,9,0,0,3,52,0,0,0,3,0,0,0,1,0,0,0,20,0,0,60,44,1,0,0,60,0,0,0,0,0,0,0,41],[17,60,15,26,0,0,30,0,44,44,26,0,0,0,32,0,0,0,9,14,0,1,0,1,0,0,24,51,0,0,1,0,0,0,0,0,0,1,20,0,0,0,9,14,0,52,47,0,0,0,24,51,0,37,10,0,0,0,1,58,1,60,3,0] >;
 

C32⋊Dic10 in GAP, Magma, Sage, TeX 

C_3^2\rtimes {\rm Dic}_{10}
 
% in TeX
 
G:=Group("C3^2:Dic10");
 
// GroupNames label
 
G:=SmallGroup(360,136);
 
// by ID
 
G=gap.SmallGroup(360,136);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-5,24,73,31,963,585,111,964,130,376,10373]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^20=1,d^2=c^10,a*b=b*a,c*a*c^-1=b,d*a*d^-1=a*b^-1,c*b*c^-1=a^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊Dic10 in TeX
Character table of C32⋊Dic10 in TeX

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