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## G = C6×C3.A4order 216 = 23·33

### Direct product of C6 and C3.A4

Aliases: C6×C3.A4, C62.6C6, (C22×C6)⋊C9, (C2×C6)⋊2C18, C3.2(C6×A4), C6.5(C3×A4), (C3×C6).5A4, C232(C3×C9), C222(C3×C18), C32.3(C2×A4), (C2×C62).1C3, (C22×C6).3C32, (C2×C6).3(C3×C6), SmallGroup(216,105)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×C3.A4
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C6×C3.A4
 Lower central C22 — C6×C3.A4
 Upper central C1 — C3×C6

Generators and relations for C6×C3.A4
G = < a,b,c,d,e | a6=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 >

Subgroups: 136 in 64 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C6, C23, C9, C32, C2×C6, C2×C6, C2×C6, C18, C3×C6, C3×C6, C22×C6, C22×C6, C3×C9, C3.A4, C62, C62, C3×C18, C2×C3.A4, C2×C62, C3×C3.A4, C6×C3.A4
Quotients: C1, C2, C3, C6, C9, C32, A4, C18, C3×C6, C2×A4, C3×C9, C3.A4, C3×A4, C3×C18, C2×C3.A4, C6×A4, C3×C3.A4, C6×C3.A4

Smallest permutation representation of C6×C3.A4
On 54 points
Generators in S54
(1 33 24 52 15 41)(2 34 25 53 16 42)(3 35 26 54 17 43)(4 36 27 46 18 44)(5 28 19 47 10 45)(6 29 20 48 11 37)(7 30 21 49 12 38)(8 31 22 50 13 39)(9 32 23 51 14 40)
(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 28)(11 29)(13 31)(14 32)(16 34)(17 35)(19 45)(20 37)(22 39)(23 40)(25 42)(26 43)
(1 52)(3 54)(4 46)(6 48)(7 49)(9 51)(11 29)(12 30)(14 32)(15 33)(17 35)(18 36)(20 37)(21 38)(23 40)(24 41)(26 43)(27 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)| (1,33,24,52,15,41)(2,34,25,53,16,42)(3,35,26,54,17,43)(4,36,27,46,18,44)(5,28,19,47,10,45)(6,29,20,48,11,37)(7,30,21,49,12,38)(8,31,22,50,13,39)(9,32,23,51,14,40), (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)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,45)(20,37)(22,39)(23,40)(25,42)(26,43), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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( (1,33,24,52,15,41)(2,34,25,53,16,42)(3,35,26,54,17,43)(4,36,27,46,18,44)(5,28,19,47,10,45)(6,29,20,48,11,37)(7,30,21,49,12,38)(8,31,22,50,13,39)(9,32,23,51,14,40), (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)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,45)(20,37)(22,39)(23,40)(25,42)(26,43), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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([[(1,33,24,52,15,41),(2,34,25,53,16,42),(3,35,26,54,17,43),(4,36,27,46,18,44),(5,28,19,47,10,45),(6,29,20,48,11,37),(7,30,21,49,12,38),(8,31,22,50,13,39),(9,32,23,51,14,40)], [(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),(10,28),(11,29),(13,31),(14,32),(16,34),(17,35),(19,45),(20,37),(22,39),(23,40),(25,42),(26,43)], [(1,52),(3,54),(4,46),(6,48),(7,49),(9,51),(11,29),(12,30),(14,32),(15,33),(17,35),(18,36),(20,37),(21,38),(23,40),(24,41),(26,43),(27,44)], [(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)]])

C6×C3.A4 is a maximal subgroup of   C62.10Dic3

72 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 6A ··· 6H 6I ··· 6X 9A ··· 9R 18A ··· 18R order 1 2 2 2 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 3 3 1 ··· 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 C9 C18 A4 C2×A4 C3.A4 C3×A4 C2×C3.A4 C6×A4 kernel C6×C3.A4 C3×C3.A4 C2×C3.A4 C2×C62 C3.A4 C62 C22×C6 C2×C6 C3×C6 C32 C6 C6 C3 C3 # reps 1 1 6 2 6 2 18 18 1 1 6 2 6 2

Matrix representation of C6×C3.A4 in GL5(𝔽19)

 12 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 1
,
 1 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(19))| [12,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C6×C3.A4 in GAP, Magma, Sage, TeX

C_6\times C_3.A_4
% in TeX

G:=Group("C6xC3.A4");
// GroupNames label

G:=SmallGroup(216,105);
// by ID

G=gap.SmallGroup(216,105);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,115,1630,2927]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=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

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