Copied to
clipboard

## G = C3×C3.A4order 108 = 22·33

### Direct product of C3 and C3.A4

Aliases: C3×C3.A4, C32.2A4, C62.1C3, (C2×C6)⋊C9, C3.2(C3×A4), C222(C3×C9), (C2×C6).3C32, SmallGroup(108,20)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C3.A4
G = < a,b,c,d,e | a3=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

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

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

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

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

C3×C3.A4 is a maximal subgroup of
C32.3S4  C62.11C32  C62.12C32  C62.16C32  He3.A4  He3⋊A4  C62.C32  3- 1+2⋊A4  C62⋊C9  A4×C3×C9  He3.2A4  C62.9C32
C3×C3.A4 is a maximal quotient of
C62.C9  C62.12C32  C62⋊C9

36 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 9A ··· 9R order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 size 1 3 1 ··· 1 3 ··· 3 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 3 3 3 type + + image C1 C3 C3 C9 A4 C3.A4 C3×A4 kernel C3×C3.A4 C3.A4 C62 C2×C6 C32 C3 C3 # reps 1 6 2 18 1 6 2

Matrix representation of C3×C3.A4 in GL4(𝔽19) generated by

 7 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 1 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 1 0 0 0 0 1 0 0 0 6 18 0 0 17 0 18
,
 1 0 0 0 0 18 0 0 0 0 18 0 0 2 0 1
,
 1 0 0 0 0 6 17 0 0 0 13 1 0 0 2 0
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,6,17,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,2,0,0,18,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,17,13,2,0,0,1,0] >;

C3×C3.A4 in GAP, Magma, Sage, TeX

C_3\times C_3.A_4
% in TeX

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

G:=SmallGroup(108,20);
// by ID

G=gap.SmallGroup(108,20);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,45,1083,2029]);
// Polycyclic

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

Export

׿
×
𝔽