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G = C9.A4order 108 = 22·33

The central extension by C9 of A4

metabelian, soluble, monomial, A-group

Aliases: C9.A4, C22⋊C27, (C2×C6).C9, (C2×C18).C3, C3.(C3.A4), SmallGroup(108,3)

Series: Derived Chief Lower central Upper central

C1C22 — C9.A4
C1C22C2×C6C2×C18 — C9.A4
C22 — C9.A4
C1C9

Generators and relations for C9.A4
 G = < a,b,c,d | a9=b2=c2=1, d3=a, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

3C2
3C6
3C18
4C27

Smallest permutation representation of C9.A4
On 54 points
Generators in S54
(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 40)(3 41)(5 43)(6 44)(8 46)(9 47)(11 49)(12 50)(14 52)(15 53)(17 28)(18 29)(20 31)(21 32)(23 34)(24 35)(26 37)(27 38)
(1 39)(3 41)(4 42)(6 44)(7 45)(9 47)(10 48)(12 50)(13 51)(15 53)(16 54)(18 29)(19 30)(21 32)(22 33)(24 35)(25 36)(27 38)
(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)| (1,4,7,10,13,16,19,22,25)(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), (2,40)(3,41)(5,43)(6,44)(8,46)(9,47)(11,49)(12,50)(14,52)(15,53)(17,28)(18,29)(20,31)(21,32)(23,34)(24,35)(26,37)(27,38), (1,39)(3,41)(4,42)(6,44)(7,45)(9,47)(10,48)(12,50)(13,51)(15,53)(16,54)(18,29)(19,30)(21,32)(22,33)(24,35)(25,36)(27,38), (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( (1,4,7,10,13,16,19,22,25)(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), (2,40)(3,41)(5,43)(6,44)(8,46)(9,47)(11,49)(12,50)(14,52)(15,53)(17,28)(18,29)(20,31)(21,32)(23,34)(24,35)(26,37)(27,38), (1,39)(3,41)(4,42)(6,44)(7,45)(9,47)(10,48)(12,50)(13,51)(15,53)(16,54)(18,29)(19,30)(21,32)(22,33)(24,35)(25,36)(27,38), (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([(1,4,7,10,13,16,19,22,25),(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)], [(2,40),(3,41),(5,43),(6,44),(8,46),(9,47),(11,49),(12,50),(14,52),(15,53),(17,28),(18,29),(20,31),(21,32),(23,34),(24,35),(26,37),(27,38)], [(1,39),(3,41),(4,42),(6,44),(7,45),(9,47),(10,48),(12,50),(13,51),(15,53),(16,54),(18,29),(19,30),(21,32),(22,33),(24,35),(25,36),(27,38)], [(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)])

C9.A4 is a maximal subgroup of   C9.S4  A4×C27  C27⋊A4  C62.C9  C42⋊C27  C24⋊C27
C9.A4 is a maximal quotient of   Q8⋊C27  C27.A4  C42⋊C27  C24⋊C27

36 conjugacy classes

class 1  2 3A3B6A6B9A···9F18A···18F27A···27R
order1233669···918···1827···27
size1311331···13···34···4

36 irreducible representations

dim1111333
type++
imageC1C3C9C27A4C3.A4C9.A4
kernelC9.A4C2×C18C2×C6C22C9C3C1
# reps12618126

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

2700
0270
0027
,
100
221080
270108
,
10800
01080
8201
,
221070
65871
11820
G:=sub<GL(3,GF(109))| [27,0,0,0,27,0,0,0,27],[1,22,27,0,108,0,0,0,108],[108,0,82,0,108,0,0,0,1],[22,65,11,107,87,82,0,1,0] >;

C9.A4 in GAP, Magma, Sage, TeX

C_9.A_4
% in TeX

G:=Group("C9.A4");
// GroupNames label

G:=SmallGroup(108,3);
// by ID

G=gap.SmallGroup(108,3);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,15,36,1083,2029]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^2=1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of C9.A4 in TeX

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