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## G = C3×C32⋊Q16order 432 = 24·33

### Direct product of C3 and C32⋊Q16

Aliases: C3×C32⋊Q16, C331Q16, C6.23S3≀C2, C32⋊(C3×Q16), C322Q8.C6, (C32×C6).5D4, C322C8.2C6, C2.5(C3×S3≀C2), (C3×C6).5(C3×D4), C3⋊Dic3.7(C2×C6), (C3×C322C8).3C2, (C3×C322Q8).2C2, (C3×C3⋊Dic3).33C22, SmallGroup(432,578)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C3×C32⋊Q16
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×C32⋊2Q8 — C3×C32⋊Q16
 Lower central C32 — C3×C6 — C3⋊Dic3 — C3×C32⋊Q16
 Upper central C1 — C6

Generators and relations for C3×C32⋊Q16
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=c, dcd-1=b-1, ece-1=b, ede-1=d-1 >

Subgroups: 300 in 72 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C3, C3 [×4], C4 [×3], C6, C6 [×4], C8, Q8 [×2], C32, C32 [×4], Dic3 [×4], C12 [×7], Q16, C3×C6, C3×C6 [×4], C24, Dic6 [×2], C3×Q8 [×2], C33, C3×Dic3 [×8], C3⋊Dic3, C3×C12 [×2], C3×Q16, C32×C6, C322C8, C322Q8 [×2], C3×Dic6 [×2], C32×Dic3 [×2], C3×C3⋊Dic3, C32⋊Q16, C3×C322C8, C3×C322Q8 [×2], C3×C32⋊Q16
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, C2×C6, Q16, C3×D4, C3×Q16, S3≀C2, C32⋊Q16, C3×S3≀C2, C3×C32⋊Q16

Smallest permutation representation of C3×C32⋊Q16
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 33 47)(3 48 34)(4 41 35)(5 36 42)(6 37 43)(7 44 38)(8 45 39)(9 30 19)(10 31 20)(11 21 32)(12 22 25)(13 26 23)(14 27 24)(15 17 28)(16 18 29)
(1 40 46)(2 47 33)(3 48 34)(4 35 41)(5 36 42)(6 43 37)(7 44 38)(8 39 45)(9 30 19)(10 20 31)(11 21 32)(12 25 22)(13 26 23)(14 24 27)(15 17 28)(16 29 18)
(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)(2,33,47)(3,48,34)(4,41,35)(5,36,42)(6,37,43)(7,44,38)(8,45,39)(9,30,19)(10,31,20)(11,21,32)(12,22,25)(13,26,23)(14,27,24)(15,17,28)(16,18,29), (1,40,46)(2,47,33)(3,48,34)(4,35,41)(5,36,42)(6,43,37)(7,44,38)(8,39,45)(9,30,19)(10,20,31)(11,21,32)(12,25,22)(13,26,23)(14,24,27)(15,17,28)(16,29,18), (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)(2,33,47)(3,48,34)(4,41,35)(5,36,42)(6,37,43)(7,44,38)(8,45,39)(9,30,19)(10,31,20)(11,21,32)(12,22,25)(13,26,23)(14,27,24)(15,17,28)(16,18,29), (1,40,46)(2,47,33)(3,48,34)(4,35,41)(5,36,42)(6,43,37)(7,44,38)(8,39,45)(9,30,19)(10,20,31)(11,21,32)(12,25,22)(13,26,23)(14,24,27)(15,17,28)(16,29,18), (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),(2,33,47),(3,48,34),(4,41,35),(5,36,42),(6,37,43),(7,44,38),(8,45,39),(9,30,19),(10,31,20),(11,21,32),(12,22,25),(13,26,23),(14,27,24),(15,17,28),(16,18,29)], [(1,40,46),(2,47,33),(3,48,34),(4,35,41),(5,36,42),(6,43,37),(7,44,38),(8,39,45),(9,30,19),(10,20,31),(11,21,32),(12,25,22),(13,26,23),(14,24,27),(15,17,28),(16,29,18)], [(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)])

45 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 4A 4B 4C 6A 6B 6C ··· 6H 8A 8B 12A ··· 12P 12Q 12R 24A 24B 24C 24D order 1 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 8 8 12 ··· 12 12 12 24 24 24 24 size 1 1 1 1 4 ··· 4 12 12 18 1 1 4 ··· 4 18 18 12 ··· 12 18 18 18 18 18 18

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - image C1 C2 C2 C3 C6 C6 D4 Q16 C3×D4 C3×Q16 S3≀C2 C32⋊Q16 C3×S3≀C2 C3×C32⋊Q16 kernel C3×C32⋊Q16 C3×C32⋊2C8 C3×C32⋊2Q8 C32⋊Q16 C32⋊2C8 C32⋊2Q8 C32×C6 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 4 4 8 8

Matrix representation of C3×C32⋊Q16 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 8 0 0 0 0 64 0 0 0 0 64 0 0 0 0 8
,
 8 0 0 0 0 64 0 0 0 0 8 0 0 0 0 64
,
 0 0 46 0 0 0 0 27 0 46 0 0 46 0 0 0
,
 46 0 0 0 0 27 0 0 0 0 0 27 0 0 27 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,64,0,0,0,0,64,0,0,0,0,8],[8,0,0,0,0,64,0,0,0,0,8,0,0,0,0,64],[0,0,0,46,0,0,46,0,46,0,0,0,0,27,0,0],[46,0,0,0,0,27,0,0,0,0,0,27,0,0,27,0] >;

C3×C32⋊Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes Q_{16}
% in TeX

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

G:=SmallGroup(432,578);
// by ID

G=gap.SmallGroup(432,578);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,168,197,176,1011,514,80,4037,3036,362,1189,1203]);
// Polycyclic

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

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