Copied to
clipboard

G = C6211Dic3order 432 = 24·33

1st semidirect product of C62 and Dic3 acting via Dic3/C3=C4

metabelian, soluble, monomial

Aliases: C6211Dic3, (C3×C62)⋊2C4, C337(C22⋊C4), C32(C62⋊C4), C222(C33⋊C4), C325(C6.D4), (C6×C3⋊S3)⋊6C4, (C2×C6)⋊2(C32⋊C4), (C2×C3⋊S3)⋊5Dic3, (C2×C3⋊S3).41D6, (C3×C3⋊S3).17D4, C6.14(C2×C32⋊C4), (C2×C33⋊C4)⋊4C2, C3⋊S3.8(C3⋊D4), (C22×C3⋊S3).6S3, C2.7(C2×C33⋊C4), (C6×C3⋊S3).43C22, (C32×C6).21(C2×C4), (C3×C6).28(C2×Dic3), (C2×C6×C3⋊S3).6C2, SmallGroup(432,641)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C6211Dic3
C1C3C33C3×C3⋊S3C6×C3⋊S3C2×C33⋊C4 — C6211Dic3
C33C32×C6 — C6211Dic3
C1C2C22

Generators and relations for C6211Dic3
 G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 904 in 152 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2 [×4], C3, C3 [×4], C4 [×2], C22, C22 [×4], S3 [×8], C6, C6 [×16], C2×C4 [×2], C23, C32, C32 [×4], Dic3 [×2], D6 [×12], C2×C6, C2×C6 [×8], C22⋊C4, C3×S3 [×8], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6 [×13], C2×Dic3 [×2], C22×S3 [×2], C22×C6, C33, C32⋊C4 [×2], S3×C6 [×12], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×4], C6.D4, C3×C3⋊S3 [×2], C3×C3⋊S3, C32×C6, C32×C6, C2×C32⋊C4 [×2], S3×C2×C6 [×2], C22×C3⋊S3, C33⋊C4 [×2], C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C3×C62, C62⋊C4, C2×C33⋊C4 [×2], C2×C6×C3⋊S3, C6211Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C32⋊C4, C6.D4, C2×C32⋊C4, C33⋊C4, C62⋊C4, C2×C33⋊C4, C6211Dic3

Permutation representations of C6211Dic3
On 24 points - transitive group 24T1286
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 6 2 4 3 5)(7 12 8 10 9 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 9 2 8 3 7)(4 12 5 11 6 10)(13 20 15 24 17 22)(14 19 16 23 18 21)
(1 24 8 13)(2 20 7 17)(3 22 9 15)(4 21 11 16)(5 23 10 14)(6 19 12 18)

G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,9,2,8,3,7)(4,12,5,11,6,10)(13,20,15,24,17,22)(14,19,16,23,18,21), (1,24,8,13)(2,20,7,17)(3,22,9,15)(4,21,11,16)(5,23,10,14)(6,19,12,18)>;

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), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,9,2,8,3,7)(4,12,5,11,6,10)(13,20,15,24,17,22)(14,19,16,23,18,21), (1,24,8,13)(2,20,7,17)(3,22,9,15)(4,21,11,16)(5,23,10,14)(6,19,12,18) );

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)], [(1,6,2,4,3,5),(7,12,8,10,9,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,9,2,8,3,7),(4,12,5,11,6,10),(13,20,15,24,17,22),(14,19,16,23,18,21)], [(1,24,8,13),(2,20,7,17),(3,22,9,15),(4,21,11,16),(5,23,10,14),(6,19,12,18)])

G:=TransitiveGroup(24,1286);

42 conjugacy classes

class 1 2A2B2C2D2E3A3B···3G4A4B4C4D6A6B6C6D···6U6V6W6X6Y
order12222233···344446666···66666
size112991824···4545454542224···418181818

42 irreducible representations

dim11111222222444444
type+++++-+-+++
imageC1C2C2C4C4S3D4Dic3D6Dic3C3⋊D4C32⋊C4C2×C32⋊C4C33⋊C4C62⋊C4C2×C33⋊C4C6211Dic3
kernelC6211Dic3C2×C33⋊C4C2×C6×C3⋊S3C6×C3⋊S3C3×C62C22×C3⋊S3C3×C3⋊S3C2×C3⋊S3C2×C3⋊S3C62C3⋊S3C2×C6C6C22C3C2C1
# reps12122121114224448

Matrix representation of C6211Dic3 in GL4(𝔽7) generated by

4245
4651
3301
0004
,
1566
5116
0060
0005
,
2406
3501
5504
6140
,
5665
2015
4354
5514
G:=sub<GL(4,GF(7))| [4,4,3,0,2,6,3,0,4,5,0,0,5,1,1,4],[1,5,0,0,5,1,0,0,6,1,6,0,6,6,0,5],[2,3,5,6,4,5,5,1,0,0,0,4,6,1,4,0],[5,2,4,5,6,0,3,5,6,1,5,1,5,5,4,4] >;

C6211Dic3 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{11}{\rm Dic}_3
% in TeX

G:=Group("C6^2:11Dic3");
// GroupNames label

G:=SmallGroup(432,641);
// by ID

G=gap.SmallGroup(432,641);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,2804,298,2693,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽