Copied to
clipboard

G = C62⋊23D6  order 432 = 24·33

4th semidirect product of C62 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, rational

Aliases: C62⋊23D6, (S3×C6)⋊6D6, C33⋊26(C2×D4), C3⋊Dic3⋊16D6, (C3×Dic3)⋊5D6, C32⋊18(S3×D4), C32⋊7D4⋊6S3, C33⋊C2⋊4D4, C3⋊3(Dic3⋊D6), C33⋊8D4⋊10C2, C33⋊7D4⋊10C2, (C3×C62)⋊5C22, (C32×C6).64C23, (C32×Dic3)⋊8C22, (C2×C6)⋊6S32, C3⋊3(D4×C3⋊S3), C6.74(C2×S32), D6⋊4(C2×C3⋊S3), (C2×C3⋊S3)⋊17D6, (C3×C3⋊D4)⋊4S3, C22⋊5(S3×C3⋊S3), C33⋊8(C2×C4)⋊9C2, C3⋊D4⋊2(C3⋊S3), (S3×C3×C6)⋊14C22, Dic3⋊2(C2×C3⋊S3), (C6×C3⋊S3)⋊11C22, (C32×C3⋊D4)⋊8C2, (C3×C32⋊7D4)⋊4C2, C6.27(C22×C3⋊S3), (C3×C3⋊Dic3)⋊6C22, (C3×C6).114(C22×S3), (C22×C33⋊C2)⋊3C2, (C2×C33⋊C2)⋊11C22, (C2×S3×C3⋊S3)⋊10C2, (C2×C6)⋊5(C2×C3⋊S3), C2.27(C2×S3×C3⋊S3), SmallGroup(432,686)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C62⋊23D6
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C2×S3×C3⋊S3 — C62⋊23D6
Lower central
C33 — C32×C6 — C62⋊23D6
Upper central
C1 — C2 — C22

Generators and relations for C62⋊23D6
 G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 3296 in 452 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3⋊D4, C3×D4, C22×S3, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×D4, S3×C32, C3×C3⋊S3, C33⋊C2, C33⋊C2, C32×C6, C32×C6, C6.D6, C3⋊D12, C3×C3⋊D4, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C32⋊7D4, D4×C32, C2×S32, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C2×C33⋊C2, C3×C62, Dic3⋊D6, D4×C3⋊S3, C33⋊8(C2×C4), C33⋊7D4, C33⋊8D4, C32×C3⋊D4, C3×C32⋊7D4, C2×S3×C3⋊S3, C22×C33⋊C2, C62⋊23D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, S3×D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, Dic3⋊D6, D4×C3⋊S3, C2×S3×C3⋊S3, C62⋊23D6

Smallest permutation representation of C62⋊23D6
►On 36 points
Generators in S36
(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)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6E6F···6V6W6X6Y6Z6AA12A12B12C12D12E
order122222223···33333446···66···6666661212121212
size1126182727542···244446182···24···412121212361212121236

51 irreducible representations

dim11111111222222224444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6D6D6S32S3×D4C2×S32Dic3⋊D6
kernelC62⋊23D6C33⋊8(C2×C4)C33⋊7D4C33⋊8D4C32×C3⋊D4C3×C32⋊7D4C2×S3×C3⋊S3C22×C33⋊C2C3×C3⋊D4C32⋊7D4C33⋊C2C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3C62C2×C6C32C6C3
# reps11111111412414154548

Matrix representation of C62⋊23D6 ►in GL8(𝔽13)

10000000
312000000
000120000
00110000
000011200
00001000
000000120
000000012
,
120000000
012000000
00010000
0012120000
000001200
000011200
00000010
00000001
,
18000000
012000000
00100000
0012120000
00000100
00001000
000000121
000000120
,
10000000
01000000
00100000
0012120000
00000100
00001000
000000120
000000121

G:=sub<GL(8,GF(13))| [1,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,8,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1] >;
 

C62⋊23D6 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_{23}D_6
 
% in TeX
 
G:=Group("C6^2:23D6");
 
// GroupNames label
 
G:=SmallGroup(432,686);
 
// by ID
 
G=gap.SmallGroup(432,686);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁