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## G = C3×C32⋊2SD16order 432 = 24·33

### Direct product of C3 and C32⋊2SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C3×C32⋊2SD16
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×D6⋊S3 — C3×C32⋊2SD16
 Lower central C32 — C3×C6 — C3⋊Dic3 — C3×C32⋊2SD16
 Upper central C1 — C6

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 2A 2B 3A 3B 3C ··· 3H 4A 4B 6A 6B 6C ··· 6H 6I ··· 6P 8A 8B 12A ··· 12H 12I 12J 24A 24B 24C 24D order 1 2 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 12 24 24 24 24 size 1 1 12 1 1 4 ··· 4 12 18 1 1 4 ··· 4 12 ··· 12 18 18 12 ··· 12 18 18 18 18 18 18

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 type + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 SD16 C3×D4 C3×SD16 S3≀C2 C32⋊2SD16 C32⋊2SD16 C3×S3≀C2 C3×C32⋊2SD16 kernel C3×C32⋊2SD16 C3×C32⋊2C8 C3×D6⋊S3 C3×C32⋊2Q8 C32⋊2SD16 C32⋊2C8 D6⋊S3 C32⋊2Q8 C32×C6 C33 C3×C6 C32 C6 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 4 4 2 2 8 8

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

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 2 5 4 2 5 4 0 5 5 4 0 3 3 0 6 2
,
 4 3 1 4 0 4 3 4 6 0 4 0 4 5 0 3
,
 1 0 1 6 6 6 4 2 3 3 5 0 0 4 3 2
,
 5 2 6 2 1 2 3 2 3 4 6 2 6 2 6 1
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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