Copied to
clipboard

G = D1223D6order 288 = 25·32

7th semidirect product of D12 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D1223D6, Dic622D6, C62.138C23, (C2×C12)⋊4D6, C3⋊D46D6, C4○D127S3, (C4×S3)⋊11D6, Dic3⋊D68C2, (C6×C12)⋊6C22, D6⋊D615C2, (S3×C12)⋊1C22, D6.4D68C2, D6.D64C2, (S3×C6).5C23, (C3×C6).13C24, C6.13(S3×C23), D12⋊S313C2, D6.6(C22×S3), (C3×D12)⋊30C22, (S3×Dic3)⋊6C22, Dic3.D615C2, D6⋊S313C22, C3⋊D1214C22, C12.151(C22×S3), (C3×C12).118C23, (C3×Dic6)⋊29C22, C322Q812C22, C3⋊Dic3.39C23, (C3×Dic3).8C23, Dic3.5(C22×S3), C6.D6.9C22, (C4×S32)⋊3C2, (C2×C4)⋊10S32, C4.98(C2×S32), C32(S3×C4○D4), C326(C2×C4○D4), C3⋊S31(C4○D4), C2.15(C22×S32), C22.12(C2×S32), (C3×C4○D12)⋊12C2, (C4×C3⋊S3)⋊13C22, (C2×S32).10C22, (C3×C3⋊D4)⋊7C22, (C2×C3⋊S3).45C23, (C2×C6).14(C22×S3), (C2×C3⋊Dic3)⋊22C22, (C22×C3⋊S3).104C22, (C2×C4×C3⋊S3)⋊6C2, SmallGroup(288,954)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D1223D6
C1C3C32C3×C6S3×C6C2×S32C4×S32 — D1223D6
C32C3×C6 — D1223D6
C1C4C2×C4

Generators and relations for D1223D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >

Subgroups: 1298 in 355 conjugacy classes, 110 normal (26 characteristic)
C1, C2, C2 [×8], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×14], C6 [×2], C6 [×9], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×6], D6 [×4], D6 [×22], C2×C6 [×2], C2×C6 [×5], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×4], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, Dic6 [×2], Dic6 [×4], C4×S3 [×4], C4×S3 [×20], D12 [×2], D12 [×4], C2×Dic3 [×7], C3⋊D4 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×7], C2×C4○D4, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], S32 [×4], S3×C6 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C2×C4 [×7], C4○D12 [×2], C4○D12 [×4], S3×D4 [×6], D42S3 [×6], S3×Q8 [×2], Q83S3 [×2], C3×C4○D4 [×2], S3×Dic3 [×4], C6.D6 [×2], D6⋊S3 [×2], C3⋊D12 [×4], C322Q8 [×2], C3×Dic6 [×2], S3×C12 [×4], C3×D12 [×2], C3×C3⋊D4 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C2×S32 [×2], C22×C3⋊S3, S3×C4○D4 [×2], D12⋊S3 [×2], Dic3.D6, D6.D6 [×2], C4×S32 [×2], D6⋊D6, D6.4D6 [×2], Dic3⋊D6 [×2], C3×C4○D12 [×2], C2×C4×C3⋊S3, D1223D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, S3×C23 [×2], C2×S32 [×3], S3×C4○D4 [×2], C22×S32, D1223D6

Permutation representations of D1223D6
On 24 points - transitive group 24T610
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 5)(2 10)(4 8)(7 11)(13 15)(14 20)(16 18)(17 23)(19 21)(22 24)

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

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

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

G:=TransitiveGroup(24,610);

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C···6G6H6I6J6K12A12B12C12D12E···12J12K12L12M12N
order12222222223334444444444666···666661212121212···1212121212
size1126666991822411266669918224···41212121222224···412121212

48 irreducible representations

dim1111111111222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6D6D6C4○D4S32C2×S32C2×S32S3×C4○D4D1223D6
kernelD1223D6D12⋊S3Dic3.D6D6.D6C4×S32D6⋊D6D6.4D6Dic3⋊D6C3×C4○D12C2×C4×C3⋊S3C4○D12Dic6C4×S3D12C3⋊D4C2×C12C3⋊S3C2×C4C4C22C3C1
# reps1212212221224242412144

Matrix representation of D1223D6 in GL4(𝔽5) generated by

4403
1042
4411
2200
,
3401
0224
1144
2021
,
3422
4043
0342
0133
,
1313
2130
2303
4203
G:=sub<GL(4,GF(5))| [4,1,4,2,4,0,4,2,0,4,1,0,3,2,1,0],[3,0,1,2,4,2,1,0,0,2,4,2,1,4,4,1],[3,4,0,0,4,0,3,1,2,4,4,3,2,3,2,3],[1,2,2,4,3,1,3,2,1,3,0,0,3,0,3,3] >;

D1223D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{23}D_6
% in TeX

G:=Group("D12:23D6");
// GroupNames label

G:=SmallGroup(288,954);
// by ID

G=gap.SmallGroup(288,954);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽