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## G = C3×C32⋊3Q16order 432 = 24·33

### Direct product of C3 and C32⋊3Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊3Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×Dic6 — C3×C32⋊3Q16
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊3Q16
 Upper central C1 — C6 — C12

Generators and relations for C3×C323Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 344 in 110 conjugacy classes, 36 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, C32, Dic3, C12, C12, Q16, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, Dic12, C3⋊Q16, C3×Q16, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, C324Q8, Q8×C32, C32×Dic3, C3×C3⋊Dic3, C32×C12, C323Q16, C3×Dic12, C3×C3⋊Q16, C32×C3⋊C8, C32×Dic6, C3×C324Q8, C3×C323Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, D12, C3⋊D4, C3×D4, S32, S3×C6, Dic12, C3⋊Q16, C3×Q16, C3⋊D12, C3×D12, C3×C3⋊D4, C3×S32, C323Q16, C3×Dic12, C3×C3⋊Q16, C3×C3⋊D12, C3×C323Q16

Smallest permutation representation of C3×C323Q16
On 48 points
Generators in S48
(1 38 25)(2 39 26)(3 40 27)(4 33 28)(5 34 29)(6 35 30)(7 36 31)(8 37 32)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)
(1 38 25)(2 26 39)(3 40 27)(4 28 33)(5 34 29)(6 30 35)(7 36 31)(8 32 37)(9 48 22)(10 23 41)(11 42 24)(12 17 43)(13 44 18)(14 19 45)(15 46 20)(16 21 47)
(1 25 38)(2 26 39)(3 27 40)(4 28 33)(5 29 34)(6 30 35)(7 31 36)(8 32 37)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 43 37 47)(34 42 38 46)(35 41 39 45)(36 48 40 44)

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

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

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

72 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 6K 8A 8B 12A ··· 12H 12I ··· 12Q 12R ··· 12Y 12Z 12AA 24A ··· 24P order 1 2 3 3 3 ··· 3 3 3 3 4 4 4 6 6 6 ··· 6 6 6 6 8 8 12 ··· 12 12 ··· 12 12 ··· 12 12 12 24 ··· 24 size 1 1 1 1 2 ··· 2 4 4 4 2 12 36 1 1 2 ··· 2 4 4 4 6 6 2 ··· 2 4 ··· 4 12 ··· 12 36 36 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 Q16 C3×S3 C3×S3 D12 C3⋊D4 C3×D4 S3×C6 Dic12 C3×Q16 C3×D12 C3×C3⋊D4 C3×Dic12 S32 C3⋊Q16 C3⋊D12 C3×S32 C32⋊3Q16 C3×C3⋊Q16 C3×C3⋊D12 C3×C32⋊3Q16 kernel C3×C32⋊3Q16 C32×C3⋊C8 C32×Dic6 C3×C32⋊4Q8 C32⋊3Q16 C3×C3⋊C8 C3×Dic6 C32⋊4Q8 C3×C3⋊C8 C3×Dic6 C32×C6 C3×C12 C33 C3⋊C8 Dic6 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×C323Q16 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 22 59 0 0 0 0 0 10 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 56 70 0 0 0 0 48 17 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,59,10,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[56,48,0,0,0,0,70,17,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×C323Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_3Q_{16}
% in TeX

G:=Group("C3xC3^2:3Q16");
// GroupNames label

G:=SmallGroup(432,424);
// by ID

G=gap.SmallGroup(432,424);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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