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## G = C3×C32⋊A4order 324 = 22·34

### Direct product of C3 and C32⋊A4

Aliases: C3×C32⋊A4, C333A4, C622C32, (C2×C6)⋊He3, C322(C3×A4), (C3×C62)⋊3C3, C222(C3×He3), (C32×A4)⋊4C3, (C3×A4)⋊2C32, C3.10(C32×A4), (C2×C6).10C33, SmallGroup(324,135)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C32⋊A4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32×A4 — C3×C32⋊A4
 Lower central C22 — C2×C6 — C3×C32⋊A4
 Upper central C1 — C32 — C33

Generators and relations for C3×C32⋊A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 520 in 140 conjugacy classes, 42 normal (9 characteristic)
C1, C2, C3, C3, C3, C22, C6, C32, C32, C32, A4, C2×C6, C2×C6, C2×C6, C3×C6, He3, C33, C33, C3×A4, C3×A4, C62, C62, C62, C32×C6, C3×He3, C32⋊A4, C32×A4, C3×C62, C3×C32⋊A4
Quotients: C1, C3, C32, A4, He3, C33, C3×A4, C3×He3, C32⋊A4, C32×A4, C3×C32⋊A4

Smallest permutation representation of C3×C32⋊A4
On 54 points
Generators in S54
(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)(49 50 51)(52 53 54)
(7 47 42)(8 48 40)(9 46 41)(13 54 50)(14 52 51)(15 53 49)(22 30 37)(23 28 38)(24 29 39)(31 35 44)(32 36 45)(33 34 43)
(1 10 20)(2 11 21)(3 12 19)(4 18 27)(5 16 25)(6 17 26)(7 42 47)(8 40 48)(9 41 46)(13 50 54)(14 51 52)(15 49 53)(22 30 37)(23 28 38)(24 29 39)(31 35 44)(32 36 45)(33 34 43)
(1 17)(2 18)(3 16)(4 21)(5 19)(6 20)(10 26)(11 27)(12 25)(22 34)(23 35)(24 36)(28 44)(29 45)(30 43)(31 38)(32 39)(33 37)
(1 17)(2 18)(3 16)(4 21)(5 19)(6 20)(7 50)(8 51)(9 49)(10 26)(11 27)(12 25)(13 47)(14 48)(15 46)(40 52)(41 53)(42 54)
(1 42 22)(2 40 23)(3 41 24)(4 51 31)(5 49 32)(6 50 33)(7 37 20)(8 38 21)(9 39 19)(10 47 30)(11 48 28)(12 46 29)(13 43 26)(14 44 27)(15 45 25)(16 53 36)(17 54 34)(18 52 35)

G:=sub<Sym(54)| (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)(49,50,51)(52,53,54), (7,47,42)(8,48,40)(9,46,41)(13,54,50)(14,52,51)(15,53,49)(22,30,37)(23,28,38)(24,29,39)(31,35,44)(32,36,45)(33,34,43), (1,10,20)(2,11,21)(3,12,19)(4,18,27)(5,16,25)(6,17,26)(7,42,47)(8,40,48)(9,41,46)(13,50,54)(14,51,52)(15,49,53)(22,30,37)(23,28,38)(24,29,39)(31,35,44)(32,36,45)(33,34,43), (1,17)(2,18)(3,16)(4,21)(5,19)(6,20)(10,26)(11,27)(12,25)(22,34)(23,35)(24,36)(28,44)(29,45)(30,43)(31,38)(32,39)(33,37), (1,17)(2,18)(3,16)(4,21)(5,19)(6,20)(7,50)(8,51)(9,49)(10,26)(11,27)(12,25)(13,47)(14,48)(15,46)(40,52)(41,53)(42,54), (1,42,22)(2,40,23)(3,41,24)(4,51,31)(5,49,32)(6,50,33)(7,37,20)(8,38,21)(9,39,19)(10,47,30)(11,48,28)(12,46,29)(13,43,26)(14,44,27)(15,45,25)(16,53,36)(17,54,34)(18,52,35)>;

G:=Group( (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)(49,50,51)(52,53,54), (7,47,42)(8,48,40)(9,46,41)(13,54,50)(14,52,51)(15,53,49)(22,30,37)(23,28,38)(24,29,39)(31,35,44)(32,36,45)(33,34,43), (1,10,20)(2,11,21)(3,12,19)(4,18,27)(5,16,25)(6,17,26)(7,42,47)(8,40,48)(9,41,46)(13,50,54)(14,51,52)(15,49,53)(22,30,37)(23,28,38)(24,29,39)(31,35,44)(32,36,45)(33,34,43), (1,17)(2,18)(3,16)(4,21)(5,19)(6,20)(10,26)(11,27)(12,25)(22,34)(23,35)(24,36)(28,44)(29,45)(30,43)(31,38)(32,39)(33,37), (1,17)(2,18)(3,16)(4,21)(5,19)(6,20)(7,50)(8,51)(9,49)(10,26)(11,27)(12,25)(13,47)(14,48)(15,46)(40,52)(41,53)(42,54), (1,42,22)(2,40,23)(3,41,24)(4,51,31)(5,49,32)(6,50,33)(7,37,20)(8,38,21)(9,39,19)(10,47,30)(11,48,28)(12,46,29)(13,43,26)(14,44,27)(15,45,25)(16,53,36)(17,54,34)(18,52,35) );

G=PermutationGroup([[(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),(49,50,51),(52,53,54)], [(7,47,42),(8,48,40),(9,46,41),(13,54,50),(14,52,51),(15,53,49),(22,30,37),(23,28,38),(24,29,39),(31,35,44),(32,36,45),(33,34,43)], [(1,10,20),(2,11,21),(3,12,19),(4,18,27),(5,16,25),(6,17,26),(7,42,47),(8,40,48),(9,41,46),(13,50,54),(14,51,52),(15,49,53),(22,30,37),(23,28,38),(24,29,39),(31,35,44),(32,36,45),(33,34,43)], [(1,17),(2,18),(3,16),(4,21),(5,19),(6,20),(10,26),(11,27),(12,25),(22,34),(23,35),(24,36),(28,44),(29,45),(30,43),(31,38),(32,39),(33,37)], [(1,17),(2,18),(3,16),(4,21),(5,19),(6,20),(7,50),(8,51),(9,49),(10,26),(11,27),(12,25),(13,47),(14,48),(15,46),(40,52),(41,53),(42,54)], [(1,42,22),(2,40,23),(3,41,24),(4,51,31),(5,49,32),(6,50,33),(7,37,20),(8,38,21),(9,39,19),(10,47,30),(11,48,28),(12,46,29),(13,43,26),(14,44,27),(15,45,25),(16,53,36),(17,54,34),(18,52,35)]])

60 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 3O ··· 3AF 6A ··· 6Z order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 size 1 3 1 ··· 1 3 ··· 3 12 ··· 12 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 3 3 3 3 type + + image C1 C3 C3 C3 A4 He3 C3×A4 C32⋊A4 kernel C3×C32⋊A4 C32⋊A4 C32×A4 C3×C62 C33 C2×C6 C32 C3 # reps 1 18 6 2 1 6 8 18

Matrix representation of C3×C32⋊A4 in GL6(𝔽7)

 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 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 2 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 6 0 0 0 0 0 0 6 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 1
,
 6 0 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 2 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 4 0 0

G:=sub<GL(6,GF(7))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,2,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,4,0] >;

C3×C32⋊A4 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes A_4
% in TeX

G:=Group("C3xC3^2:A4");
// GroupNames label

G:=SmallGroup(324,135);
// by ID

G=gap.SmallGroup(324,135);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,650,4864,8753]);
// Polycyclic

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

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