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## G = C3×C32⋊2C16order 432 = 24·33

### Direct product of C3 and C32⋊2C16

Aliases: C3×C322C16, C333C16, C323C48, (C3×C6).3C24, (C3×C12).7C12, (C32×C6).3C8, (C32×C12).2C4, C324C8.4C6, C12.10(C32⋊C4), C6.4(C322C8), C4.2(C3×C32⋊C4), C2.(C3×C322C8), (C3×C324C8).1C2, SmallGroup(432,412)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C32⋊2C16
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C3×C32⋊4C8 — C3×C32⋊2C16
 Lower central C32 — C3×C32⋊2C16
 Upper central C1 — C12

Generators and relations for C3×C322C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Smallest permutation representation of C3×C322C16
On 48 points
Generators in S48
(1 43 27)(2 44 28)(3 45 29)(4 46 30)(5 47 31)(6 48 32)(7 33 17)(8 34 18)(9 35 19)(10 36 20)(11 37 21)(12 38 22)(13 39 23)(14 40 24)(15 41 25)(16 42 26)
(2 44 28)(4 30 46)(6 48 32)(8 18 34)(10 36 20)(12 22 38)(14 40 24)(16 26 42)
(1 43 27)(2 44 28)(3 29 45)(4 30 46)(5 47 31)(6 48 32)(7 17 33)(8 18 34)(9 35 19)(10 36 20)(11 21 37)(12 22 38)(13 39 23)(14 40 24)(15 25 41)(16 26 42)
(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)

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

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

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

72 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 4A 4B 6A 6B 6C ··· 6H 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12P 16A ··· 16H 24A ··· 24H 48A ··· 48P order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 4 ··· 4 1 1 1 1 4 ··· 4 9 9 9 9 1 1 1 1 4 ··· 4 9 ··· 9 9 ··· 9 9 ··· 9

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C48 C32⋊C4 C32⋊2C8 C3×C32⋊C4 C32⋊2C16 C3×C32⋊2C8 C3×C32⋊2C16 kernel C3×C32⋊2C16 C3×C32⋊4C8 C32⋊2C16 C32×C12 C32⋊4C8 C32×C6 C3×C12 C33 C3×C6 C32 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 2 2 4 4 4 8

Matrix representation of C3×C322C16 in GL4(𝔽97) generated by

 61 0 0 0 0 61 0 0 0 0 61 0 0 0 0 61
,
 1 0 0 12 0 1 0 18 0 0 35 73 0 0 0 61
,
 61 0 0 0 0 35 0 66 0 0 35 73 0 0 0 61
,
 27 0 1 0 89 0 0 0 47 1 0 0 38 0 0 70
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[1,0,0,0,0,1,0,0,0,0,35,0,12,18,73,61],[61,0,0,0,0,35,0,0,0,0,35,0,0,66,73,61],[27,89,47,38,0,0,1,0,1,0,0,0,0,0,0,70] >;

C3×C322C16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2C_{16}
% in TeX

G:=Group("C3xC3^2:2C16");
// GroupNames label

G:=SmallGroup(432,412);
// by ID

G=gap.SmallGroup(432,412);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-3,3,42,58,80,14117,691,18822,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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