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## G = C3×C3⋊D24order 432 = 24·33

### Direct product of C3 and C3⋊D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C3⋊D24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×D12 — C3×C3⋊D24
 Lower central C32 — C3×C6 — C3×C12 — C3×C3⋊D24
 Upper central C1 — C6 — C12

Generators and relations for C3×C3⋊D24
G = < a,b,c,d | a3=b3=c24=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 616 in 134 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, D12, C3×D4, C33, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, D24, D4⋊S3, C3×D8, S3×C32, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C3×C24, C3×D12, C3×D12, C12⋊S3, D4×C32, C32×C12, S3×C3×C6, C6×C3⋊S3, C3⋊D24, C3×D24, C3×D4⋊S3, C32×C3⋊C8, C32×D12, C3×C12⋊S3, C3×C3⋊D24
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, D12, C3⋊D4, C3×D4, S32, S3×C6, D24, D4⋊S3, C3×D8, C3⋊D12, C3×D12, C3×C3⋊D4, C3×S32, C3⋊D24, C3×D24, C3×D4⋊S3, C3×C3⋊D12, C3×C3⋊D24

Smallest permutation representation of C3×C3⋊D24
On 48 points
Generators in S48
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 17 9)(2 10 18)(3 19 11)(4 12 20)(5 21 13)(6 14 22)(7 23 15)(8 16 24)(25 33 41)(26 42 34)(27 35 43)(28 44 36)(29 37 45)(30 46 38)(31 39 47)(32 48 40)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 48)(22 47)(23 46)(24 45)

G:=sub<Sym(48)| (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,17,9)(2,10,18)(3,19,11)(4,12,20)(5,21,13)(6,14,22)(7,23,15)(8,16,24)(25,33,41)(26,42,34)(27,35,43)(28,44,36)(29,37,45)(30,46,38)(31,39,47)(32,48,40), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,48)(22,47)(23,46)(24,45)>;

G:=Group( (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,17,9)(2,10,18)(3,19,11)(4,12,20)(5,21,13)(6,14,22)(7,23,15)(8,16,24)(25,33,41)(26,42,34)(27,35,43)(28,44,36)(29,37,45)(30,46,38)(31,39,47)(32,48,40), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,48)(22,47)(23,46)(24,45) );

G=PermutationGroup([[(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48)], [(1,17,9),(2,10,18),(3,19,11),(4,12,20),(5,21,13),(6,14,22),(7,23,15),(8,16,24),(25,33,41),(26,42,34),(27,35,43),(28,44,36),(29,37,45),(30,46,38),(31,39,47),(32,48,40)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,48),(22,47),(23,46),(24,45)]])

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6S 6T 6U 8A 8B 12A ··· 12H 12I ··· 12Q 24A ··· 24P order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 6 6 6 ··· 6 6 6 6 6 ··· 6 6 6 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 12 36 1 1 2 ··· 2 4 4 4 2 1 1 2 ··· 2 4 4 4 12 ··· 12 36 36 6 6 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 S3 D4 D6 D8 C3×S3 C3×S3 D12 C3⋊D4 C3×D4 S3×C6 D24 C3×D8 C3×D12 C3×C3⋊D4 C3×D24 S32 D4⋊S3 C3⋊D12 C3×S32 C3⋊D24 C3×D4⋊S3 C3×C3⋊D12 C3×C3⋊D24 kernel C3×C3⋊D24 C32×C3⋊C8 C32×D12 C3×C12⋊S3 C3⋊D24 C3×C3⋊C8 C3×D12 C12⋊S3 C3×C3⋊C8 C3×D12 C32×C6 C3×C12 C33 C3⋊C8 D12 C3×C6 C3×C6 C3×C6 C12 C32 C32 C6 C6 C3 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 1 1 1 2 2 2 2 4

Matrix representation of C3×C3⋊D24 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 32 0 0 0 0 57 32 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 32 0 0 0 0 16 0 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,57,0,0,0,0,32,32,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,16,0,0,0,0,32,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C3⋊D24 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes D_{24}
% in TeX

G:=Group("C3xC3:D24");
// GroupNames label

G:=SmallGroup(432,419);
// by ID

G=gap.SmallGroup(432,419);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,260,1011,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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