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## G = C3×C62.C22order 432 = 24·33

### Direct product of C3 and C62.C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C62.C22
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — Dic3×C3×C6 — C3×C62.C22
 Lower central C32 — C3×C6 — C3×C62.C22
 Upper central C1 — C2×C6

Generators and relations for C3×C62.C22
G = < a,b,c,d,e | a3=b6=c6=1, d2=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=c3d >

Subgroups: 448 in 162 conjugacy classes, 56 normal (24 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C3×C4⋊C4, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C3×C62, C62.C22, C3×Dic3⋊C4, Dic3×C3×C6, C6×C3⋊Dic3, C3×C62.C22
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, S32, S3×C6, Dic3⋊C4, C3×C4⋊C4, C6.D6, D6⋊S3, C322Q8, C3×Dic6, S3×C12, C3×C3⋊D4, C3×S32, C62.C22, C3×Dic3⋊C4, C3×C6.D6, C3×D6⋊S3, C3×C322Q8, C3×C62.C22

Smallest permutation representation of C3×C62.C22
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(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 14 5 18 3 16)(2 15 6 13 4 17)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 26 21 28 23 30)(20 27 22 29 24 25)(31 41 33 37 35 39)(32 42 34 38 36 40)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 29 16 26)(14 28 17 25)(15 27 18 30)(37 44 40 47)(38 43 41 46)(39 48 42 45)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 12A ··· 12AF 12AG 12AH 12AI 12AJ order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 1 1 2 ··· 2 4 4 4 6 6 6 6 18 18 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - + - + + - - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 D6 C3×S3 Dic6 C4×S3 C3⋊D4 C3×D4 C3×Q8 S3×C6 C3×Dic6 S3×C12 C3×C3⋊D4 S32 C6.D6 D6⋊S3 C32⋊2Q8 C3×S32 C3×C6.D6 C3×D6⋊S3 C3×C32⋊2Q8 kernel C3×C62.C22 Dic3×C3×C6 C6×C3⋊Dic3 C62.C22 C3×C3⋊Dic3 C6×Dic3 C2×C3⋊Dic3 C3⋊Dic3 C6×Dic3 C32×C6 C32×C6 C62 C2×Dic3 C3×C6 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 2 1 1 2 4 4 4 4 2 2 4 8 8 8 1 1 1 1 2 2 2 2

Matrix representation of C3×C62.C22 in GL8(𝔽13)

 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 10 6 0 0 0 0 0 0 7 3 0 0 0 0 0 0 0 0 11 4 0 0 0 0 0 0 9 2 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[10,7,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3×C62.C22 in GAP, Magma, Sage, TeX

C_3\times C_6^2.C_2^2
% in TeX

G:=Group("C3xC6^2.C2^2");
// GroupNames label

G:=SmallGroup(432,429);
// by ID

G=gap.SmallGroup(432,429);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,92,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=c^6=1,d^2=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^3*d>;
// generators/relations

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