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G = C3⋊S33C16order 288 = 25·32

2nd semidirect product of C3⋊S3 and C16 acting via C16/C8=C2

metabelian, soluble, monomial, A-group

Aliases: C3⋊S33C16, (C3×C24).2C4, C323(C2×C16), C8.5(C32⋊C4), C3⋊Dic3.6C8, C322C165C2, C324C8.32C22, (C2×C3⋊S3).6C8, (C8×C3⋊S3).6C2, (C3×C6).8(C2×C8), (C4×C3⋊S3).10C4, (C3×C12).8(C2×C4), C4.15(C2×C32⋊C4), C2.1(C3⋊S33C8), SmallGroup(288,412)

Series: Derived Chief Lower central Upper central

C1C32 — C3⋊S33C16
C1C32C3×C6C3×C12C324C8C322C16 — C3⋊S33C16
C32 — C3⋊S33C16
C1C8

Generators and relations for C3⋊S33C16
 G = < a,b,c,d | a3=b3=c2=d16=1, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a-1b-1, cd=dc >

9C2
9C2
2C3
2C3
9C22
9C4
2C6
2C6
6S3
6S3
6S3
6S3
9C8
9C2×C4
2C12
2C12
6Dic3
6D6
6Dic3
6D6
9C2×C8
9C16
9C16
2C24
2C24
6C3⋊C8
6C4×S3
6C4×S3
6C3⋊C8
9C2×C16
6S3×C8
6S3×C8

Smallest permutation representation of C3⋊S33C16
On 48 points
Generators in S48
(1 20 43)(2 21 44)(3 45 22)(4 46 23)(5 24 47)(6 25 48)(7 33 26)(8 34 27)(9 28 35)(10 29 36)(11 37 30)(12 38 31)(13 32 39)(14 17 40)(15 41 18)(16 42 19)
(2 44 21)(4 23 46)(6 48 25)(8 27 34)(10 36 29)(12 31 38)(14 40 17)(16 19 42)
(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(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)| (1,20,43)(2,21,44)(3,45,22)(4,46,23)(5,24,47)(6,25,48)(7,33,26)(8,34,27)(9,28,35)(10,29,36)(11,37,30)(12,38,31)(13,32,39)(14,17,40)(15,41,18)(16,42,19), (2,44,21)(4,23,46)(6,48,25)(8,27,34)(10,36,29)(12,31,38)(14,40,17)(16,19,42), (17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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( (1,20,43)(2,21,44)(3,45,22)(4,46,23)(5,24,47)(6,25,48)(7,33,26)(8,34,27)(9,28,35)(10,29,36)(11,37,30)(12,38,31)(13,32,39)(14,17,40)(15,41,18)(16,42,19), (2,44,21)(4,23,46)(6,48,25)(8,27,34)(10,36,29)(12,31,38)(14,40,17)(16,19,42), (17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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([(1,20,43),(2,21,44),(3,45,22),(4,46,23),(5,24,47),(6,25,48),(7,33,26),(8,34,27),(9,28,35),(10,29,36),(11,37,30),(12,38,31),(13,32,39),(14,17,40),(15,41,18),(16,42,19)], [(2,44,21),(4,23,46),(6,48,25),(8,27,34),(10,36,29),(12,31,38),(14,40,17),(16,19,42)], [(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(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)])

48 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B8A8B8C8D8E8F8G8H12A12B12C12D16A···16P24A···24H
order122233444466888888881212121216···1624···24
size1199441199441111999944449···94···4

48 irreducible representations

dim111111114444
type+++++
imageC1C2C2C4C4C8C8C16C32⋊C4C2×C32⋊C4C3⋊S33C8C3⋊S33C16
kernelC3⋊S33C16C322C16C8×C3⋊S3C3×C24C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C3⋊S3C8C4C2C1
# reps1212244162248

Matrix representation of C3⋊S33C16 in GL4(𝔽97) generated by

01064
9696330
00096
00196
,
10640
01640
00961
00960
,
01064
10064
00196
00096
,
2208182
2208181
66337575
336400
G:=sub<GL(4,GF(97))| [0,96,0,0,1,96,0,0,0,33,0,1,64,0,96,96],[1,0,0,0,0,1,0,0,64,64,96,96,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,64,64,96,96],[22,22,66,33,0,0,33,64,81,81,75,0,82,81,75,0] >;

C3⋊S33C16 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\rtimes_3C_{16}
% in TeX

G:=Group("C3:S3:3C16");
// GroupNames label

G:=SmallGroup(288,412);
// by ID

G=gap.SmallGroup(288,412);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,58,80,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^2=d^16=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,c*d=d*c>;
// generators/relations

Export

Subgroup lattice of C3⋊S33C16 in TeX

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