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## G = C2×C9.A4order 216 = 23·33

### Direct product of C2 and C9.A4

Aliases: C2×C9.A4, C23⋊C27, C22⋊C54, C18.3A4, C9.(C2×A4), (C2×C6).C18, (C22×C6).C9, (C2×C18).2C6, (C22×C18).C3, C6.2(C3.A4), C3.(C2×C3.A4), SmallGroup(216,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C9.A4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9.A4 — C2×C9.A4
 Lower central C22 — C2×C9.A4
 Upper central C1 — C18

Generators and relations for C2×C9.A4
G = < a,b,c,d,e | a2=b9=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 C2×C9.A4
On 54 points
Generators in S54
(1 52)(2 53)(3 54)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)
(2 53)(3 54)(5 29)(6 30)(8 32)(9 33)(11 35)(12 36)(14 38)(15 39)(17 41)(18 42)(20 44)(21 45)(23 47)(24 48)(26 50)(27 51)
(1 52)(3 54)(4 28)(6 30)(7 31)(9 33)(10 34)(12 36)(13 37)(15 39)(16 40)(18 42)(19 43)(21 45)(22 46)(24 48)(25 49)(27 51)
(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)| (1,52)(2,53)(3,54)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (2,53)(3,54)(5,29)(6,30)(8,32)(9,33)(11,35)(12,36)(14,38)(15,39)(17,41)(18,42)(20,44)(21,45)(23,47)(24,48)(26,50)(27,51), (1,52)(3,54)(4,28)(6,30)(7,31)(9,33)(10,34)(12,36)(13,37)(15,39)(16,40)(18,42)(19,43)(21,45)(22,46)(24,48)(25,49)(27,51), (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( (1,52)(2,53)(3,54)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (2,53)(3,54)(5,29)(6,30)(8,32)(9,33)(11,35)(12,36)(14,38)(15,39)(17,41)(18,42)(20,44)(21,45)(23,47)(24,48)(26,50)(27,51), (1,52)(3,54)(4,28)(6,30)(7,31)(9,33)(10,34)(12,36)(13,37)(15,39)(16,40)(18,42)(19,43)(21,45)(22,46)(24,48)(25,49)(27,51), (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([[(1,52),(2,53),(3,54),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51)], [(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54)], [(2,53),(3,54),(5,29),(6,30),(8,32),(9,33),(11,35),(12,36),(14,38),(15,39),(17,41),(18,42),(20,44),(21,45),(23,47),(24,48),(26,50),(27,51)], [(1,52),(3,54),(4,28),(6,30),(7,31),(9,33),(10,34),(12,36),(13,37),(15,39),(16,40),(18,42),(19,43),(21,45),(22,46),(24,48),(25,49),(27,51)], [(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)]])

C2×C9.A4 is a maximal subgroup of   C18.S4
C2×C9.A4 is a maximal quotient of   Q8.C54

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 9A ··· 9F 18A ··· 18F 18G ··· 18R 27A ··· 27R 54A ··· 54R order 1 2 2 2 3 3 6 6 6 6 6 6 9 ··· 9 18 ··· 18 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 3 3 1 1 1 1 3 3 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 C6 C9 C18 C27 C54 A4 C2×A4 C3.A4 C2×C3.A4 C9.A4 C2×C9.A4 kernel C2×C9.A4 C9.A4 C22×C18 C2×C18 C22×C6 C2×C6 C23 C22 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 6 6 18 18 1 1 2 2 6 6

Matrix representation of C2×C9.A4 in GL3(𝔽109) generated by

 108 0 0 0 108 0 0 0 108
,
 27 0 0 0 27 0 0 0 27
,
 1 0 0 16 108 0 22 0 108
,
 108 0 0 0 108 0 87 0 1
,
 16 107 0 8 93 1 108 87 0
G:=sub<GL(3,GF(109))| [108,0,0,0,108,0,0,0,108],[27,0,0,0,27,0,0,0,27],[1,16,22,0,108,0,0,0,108],[108,0,87,0,108,0,0,0,1],[16,8,108,107,93,87,0,1,0] >;

C2×C9.A4 in GAP, Magma, Sage, TeX

C_2\times C_9.A_4
% in TeX

G:=Group("C2xC9.A4");
// GroupNames label

G:=SmallGroup(216,22);
// by ID

G=gap.SmallGroup(216,22);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,43,68,1630,2927]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=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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