Copied to
clipboard

G = C32⋊2+ 1+4order 288 = 25·32

The semidirect product of C32 and 2+ 1+4 acting via 2+ 1+4/C23=C22

metabelian, supersoluble, monomial

Aliases: C3262+ 1+4, C62.148C23, C233S32, C3⋊D410D6, (C22×C6)⋊9D6, Dic3⋊D67C2, (C2×Dic3)⋊8D6, (C22×S3)⋊8D6, C34(D46D6), D6.4D67C2, D6.3D66C2, (C3×C6).37C24, C6.37(S3×C23), (C2×C62)⋊8C22, D6⋊S37C22, C3⋊D127C22, (S3×C6).18C23, (S3×Dic3)⋊4C22, D6.18(C22×S3), C6.D64C22, C322Q86C22, (C6×Dic3)⋊10C22, C327D410C22, C3⋊Dic3.28C23, Dic3.17(C22×S3), (C3×Dic3).19C23, (C2×S32)⋊7C22, (C6×C3⋊D4)⋊8C2, (S3×C3⋊D4)⋊6C2, (C2×C3⋊D4)⋊12S3, (S3×C2×C6)⋊13C22, C22.10(C2×S32), C2.37(C22×S32), (C2×C3⋊S3).31C23, (C2×C327D4)⋊17C2, (C3×C3⋊D4)⋊14C22, (C22×C3⋊S3)⋊10C22, (C2×C6).163(C22×S3), (C2×C3⋊Dic3)⋊13C22, SmallGroup(288,978)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C32⋊2+ 1+4
C1C3C32C3×C6S3×C6C2×S32S3×C3⋊D4 — C32⋊2+ 1+4
C32C3×C6 — C32⋊2+ 1+4
C1C2C23

Generators and relations for C32⋊2+ 1+4
 G = < a,b,c,d,e,f | a3=b3=c4=d2=f2=1, e2=c2, ab=ba, cac-1=dad=a-1, ae=ea, af=fa, bc=cb, bd=db, ebe-1=b-1, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >

Subgroups: 1410 in 359 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2 [×9], C3 [×2], C3, C4 [×6], C22, C22 [×2], C22 [×12], S3 [×10], C6 [×2], C6 [×15], C2×C4 [×9], D4 [×18], Q8 [×2], C23, C23 [×5], C32, Dic3 [×4], Dic3 [×6], C12 [×4], D6 [×4], D6 [×16], C2×C6 [×6], C2×C6 [×13], C2×D4 [×9], C4○D4 [×6], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×3], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C2×Dic3 [×7], C3⋊D4 [×8], C3⋊D4 [×20], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×S3 [×7], C22×C6 [×2], C22×C6 [×3], 2+ 1+4, C3×Dic3 [×4], C3⋊Dic3 [×2], S32 [×2], S3×C6 [×4], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3, C62, C62 [×2], C62, C4○D12 [×4], S3×D4 [×8], D42S3 [×8], C2×C3⋊D4 [×2], C2×C3⋊D4 [×7], C6×D4 [×2], S3×Dic3 [×4], C6.D6 [×2], D6⋊S3 [×2], C3⋊D12 [×4], C322Q8 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×8], C2×C3⋊Dic3, C327D4 [×4], C2×S32 [×2], S3×C2×C6 [×2], C22×C3⋊S3, C2×C62, D46D6 [×2], D6.3D6 [×4], D6.4D6 [×2], S3×C3⋊D4 [×4], Dic3⋊D6 [×2], C6×C3⋊D4 [×2], C2×C327D4, C32⋊2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2+ 1+4, S32, S3×C23 [×2], C2×S32 [×3], D46D6 [×2], C22×S32, C32⋊2+ 1+4

Permutation representations of C32⋊2+ 1+4
On 24 points - transitive group 24T582
Generators in S24
(1 16 17)(2 18 13)(3 14 19)(4 20 15)(5 23 10)(6 11 24)(7 21 12)(8 9 22)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)
(1 22 3 24)(2 23 4 21)(5 15 7 13)(6 16 8 14)(9 19 11 17)(10 20 12 18)
(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)

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

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

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

G:=TransitiveGroup(24,582);

On 24 points - transitive group 24T609
Generators in S24
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 10 14)(6 15 11)(7 12 16)(8 13 9)
(1 21 20)(2 22 17)(3 23 18)(4 24 19)(5 14 10)(6 15 11)(7 16 12)(8 13 9)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
(1 2 3 4)(5 8 7 6)(9 16 11 14)(10 13 12 15)(17 23 19 21)(18 24 20 22)
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)

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

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

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

G:=TransitiveGroup(24,609);

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C4D4E4F6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222222223334444446···66···6666612121212
size1122266661818224666618182···24···41212121212121212

45 irreducible representations

dim11111112222244444
type+++++++++++++++
imageC1C2C2C2C2C2C2S3D6D6D6D62+ 1+4S32C2×S32D46D6C32⋊2+ 1+4
kernelC32⋊2+ 1+4D6.3D6D6.4D6S3×C3⋊D4Dic3⋊D6C6×C3⋊D4C2×C327D4C2×C3⋊D4C2×Dic3C3⋊D4C22×S3C22×C6C32C23C22C3C1
# reps14242212282211344

Matrix representation of C32⋊2+ 1+4 in GL4(𝔽7) generated by

4121
3364
6642
6141
,
6031
5360
1645
3326
,
4600
3300
3214
0436
,
1420
5331
4605
3323
,
4334
3641
3034
0341
,
3031
5060
1615
3323
G:=sub<GL(4,GF(7))| [4,3,6,6,1,3,6,1,2,6,4,4,1,4,2,1],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[4,3,3,0,6,3,2,4,0,0,1,3,0,0,4,6],[1,5,4,3,4,3,6,3,2,3,0,2,0,1,5,3],[4,3,3,0,3,6,0,3,3,4,3,4,4,1,4,1],[3,5,1,3,0,0,6,3,3,6,1,2,1,0,5,3] >;

C32⋊2+ 1+4 in GAP, Magma, Sage, TeX

C_3^2\rtimes 2_+^{1+4}
% in TeX

G:=Group("C3^2:ES+(2,2)");
// GroupNames label

G:=SmallGroup(288,978);
// by ID

G=gap.SmallGroup(288,978);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=d^2=f^2=1,e^2=c^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c^2*e>;
// generators/relations

׿
×
𝔽