Copied to
clipboard

## 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, C4, C6, C6, C8, Q8, C32, C32, Dic3, C12, Q16, C3×C6, C3×C6, C24, Dic6, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×Q16, C32×C6, C322C8, C322Q8, C3×Dic6, C32×Dic3, C3×C3⋊Dic3, C32⋊Q16, C3×C322C8, C3×C322Q8, C3×C32⋊Q16
Quotients: C1, C2, C3, C22, C6, 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 11 46)(2 12 47)(3 13 48)(4 14 41)(5 15 42)(6 16 43)(7 9 44)(8 10 45)(17 34 28)(18 35 29)(19 36 30)(20 37 31)(21 38 32)(22 39 25)(23 40 26)(24 33 27)
(1 11 46)(2 12 47)(3 48 13)(4 41 14)(5 15 42)(6 16 43)(7 44 9)(8 45 10)(17 28 34)(18 29 35)(19 36 30)(20 37 31)(21 32 38)(22 25 39)(23 40 26)(24 33 27)
(1 11 46)(2 47 12)(3 48 13)(4 14 41)(5 15 42)(6 43 16)(7 44 9)(8 10 45)(17 28 34)(18 35 29)(19 36 30)(20 31 37)(21 32 38)(22 39 25)(23 40 26)(24 27 33)
(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 34 13 38)(10 33 14 37)(11 40 15 36)(12 39 16 35)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)

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

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

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

׿
×
𝔽