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G = C3×C322SD16order 432 = 24·33

Direct product of C3 and C322SD16

direct product, non-abelian, soluble, monomial

Aliases: C3×C322SD16, C335SD16, C6.22S3≀C2, D6⋊S3.C6, C322C82C6, C322Q81C6, (C32×C6).4D4, C322(C3×SD16), C2.4(C3×S3≀C2), (C3×C6).4(C3×D4), (C3×C322C8)⋊8C2, C3⋊Dic3.6(C2×C6), (C3×D6⋊S3).2C2, (C3×C322Q8)⋊12C2, (C3×C3⋊Dic3).32C22, SmallGroup(432,577)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C3×C322SD16
C1C32C3×C6C3⋊Dic3C3×C3⋊Dic3C3×D6⋊S3 — C3×C322SD16
C32C3×C6C3⋊Dic3 — C3×C322SD16
C1C6

Generators and relations for C3×C322SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c-1, ebe=b-1, dcd-1=b, ce=ec, ede=d3 >

Subgroups: 396 in 84 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, S3, C6, C6 [×8], C8, D4, Q8, C32, C32 [×4], Dic3 [×3], C12 [×4], D6, C2×C6 [×3], SD16, C3×S3 [×4], C3×C6, C3×C6 [×5], C24, Dic6, C3⋊D4, C3×D4, C3×Q8, C33, C3×Dic3 [×5], C3⋊Dic3, C3×C12, S3×C6 [×3], C62, C3×SD16, S3×C32, C32×C6, C322C8, D6⋊S3, C322Q8, C3×Dic6, C3×C3⋊D4, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C322SD16, C3×C322C8, C3×D6⋊S3, C3×C322Q8, C3×C322SD16
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, C2×C6, SD16, C3×D4, C3×SD16, S3≀C2, C322SD16, C3×S3≀C2, C3×C322SD16

Permutation representations of C3×C322SD16
On 24 points - transitive group 24T1319
Generators in S24
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 17 15)(6 18 16)(7 19 9)(8 20 10)
(2 22 12)(4 14 24)(6 18 16)(8 10 20)
(1 11 21)(3 23 13)(5 15 17)(7 19 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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,1319);

45 conjugacy classes

class 1 2A2B3A3B3C···3H4A4B6A6B6C···6H6I···6P8A8B12A···12H12I12J24A24B24C24D
order122333···344666···66···68812···12121224242424
size1112114···41218114···412···12181812···12181818181818

45 irreducible representations

dim11111111222244444
type++++++-
imageC1C2C2C2C3C6C6C6D4SD16C3×D4C3×SD16S3≀C2C322SD16C322SD16C3×S3≀C2C3×C322SD16
kernelC3×C322SD16C3×C322C8C3×D6⋊S3C3×C322Q8C322SD16C322C8D6⋊S3C322Q8C32×C6C33C3×C6C32C6C3C3C2C1
# reps11112222122442288

Matrix representation of C3×C322SD16 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
2542
5405
5403
3062
,
4314
0434
6040
4503
,
1016
6642
3350
0432
,
5262
1232
3462
6261
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,5,5,3,5,4,4,0,4,0,0,6,2,5,3,2],[4,0,6,4,3,4,0,5,1,3,4,0,4,4,0,3],[1,6,3,0,0,6,3,4,1,4,5,3,6,2,0,2],[5,1,3,6,2,2,4,2,6,3,6,6,2,2,2,1] >;

C3×C322SD16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("C3xC3^2:2SD16");
// GroupNames label

G:=SmallGroup(432,577);
// by ID

G=gap.SmallGroup(432,577);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,197,176,1011,514,80,4037,3036,362,1189,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c^-1,e*b*e=b^-1,d*c*d^-1=b,c*e=e*c,e*d*e=d^3>;
// generators/relations

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