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## G = C3×C32⋊2Q16order 432 = 24·33

### Direct product of C3 and C32⋊2Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊2Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×Dic6 — C3×C32⋊2Q16
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊2Q16
 Upper central C1 — C6 — C12

Generators and relations for C3×C322Q16
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=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 304 in 110 conjugacy classes, 36 normal (16 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C32×C6, C3×C3⋊C8, C324C8, C3×Dic6, C3×Dic6, Q8×C32, C32×Dic3, C32×C12, C322Q16, C3×C3⋊Q16, C3×C324C8, C32×Dic6, C3×C322Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊D4, C3×D4, S32, S3×C6, C3⋊Q16, C3×Q16, D6⋊S3, C3×C3⋊D4, C3×S32, C322Q16, C3×C3⋊Q16, C3×D6⋊S3, C3×C322Q16

Smallest permutation representation of C3×C322Q16
On 48 points
Generators in S48
(1 40 46)(2 33 47)(3 34 48)(4 35 41)(5 36 42)(6 37 43)(7 38 44)(8 39 45)(9 30 19)(10 31 20)(11 32 21)(12 25 22)(13 26 23)(14 27 24)(15 28 17)(16 29 18)
(1 40 46)(2 47 33)(3 34 48)(4 41 35)(5 36 42)(6 43 37)(7 38 44)(8 45 39)(9 19 30)(10 31 20)(11 21 32)(12 25 22)(13 23 26)(14 27 24)(15 17 28)(16 29 18)
(1 40 46)(2 47 33)(3 34 48)(4 41 35)(5 36 42)(6 43 37)(7 38 44)(8 45 39)(9 30 19)(10 20 31)(11 32 21)(12 22 25)(13 26 23)(14 24 27)(15 28 17)(16 18 29)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 40 13 36)(10 39 14 35)(11 38 15 34)(12 37 16 33)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)

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

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

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

63 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 12B 12C ··· 12N 12O ··· 12AD 24A 24B 24C 24D 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 24 24 24 24 size 1 1 1 1 2 ··· 2 4 4 4 2 12 12 1 1 2 ··· 2 4 4 4 18 18 2 2 4 ··· 4 12 ··· 12 18 18 18 18

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + - + - - image C1 C2 C2 C3 C6 C6 S3 D4 D6 Q16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×Q16 C3×C3⋊D4 S32 C3⋊Q16 D6⋊S3 C3×S32 C32⋊2Q16 C3×C3⋊Q16 C3×D6⋊S3 C3×C32⋊2Q16 kernel C3×C32⋊2Q16 C3×C32⋊4C8 C32×Dic6 C32⋊2Q16 C32⋊4C8 C3×Dic6 C3×Dic6 C32×C6 C3×C12 C33 Dic6 C3×C6 C3×C6 C12 C32 C6 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 2 1 2 2 4 4 2 4 4 8 1 2 1 2 2 4 2 4

Matrix representation of C3×C322Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 64 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 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 22 0 0 0 0 0 0 10 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 72 0 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

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C322Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2Q_{16}
% in TeX

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

G:=SmallGroup(432,423);
// by ID

G=gap.SmallGroup(432,423);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,2028,14118]);
// 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=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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