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G = C2×C27⋊C3order 162 = 2·34

Direct product of C2 and C27⋊C3

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C2×C27⋊C3, C54⋊C3, C18.C9, C9.C18, C272C6, C32.C18, C18.2C32, (C3×C6).C9, C9.1(C3×C6), (C3×C9).4C6, C6.3(C3×C9), (C3×C18).3C3, C3.3(C3×C18), SmallGroup(162,27)

Series: Derived Chief Lower central Upper central

C1C3 — C2×C27⋊C3
C1C3C9C3×C9C27⋊C3 — C2×C27⋊C3
C1C3 — C2×C27⋊C3
C1C18 — C2×C27⋊C3

Generators and relations for C2×C27⋊C3
 G = < a,b,c | a2=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >

3C3
3C6

Smallest permutation representation of C2×C27⋊C3
On 54 points
Generators in S54
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)

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

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

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

C2×C27⋊C3 is a maximal subgroup of   C27⋊C12

66 conjugacy classes

class 1  2 3A3B3C3D6A6B6C6D9A···9F9G9H9I9J18A···18F18G18H18I18J27A···27R54A···54R
order12333366669···9999918···181818181827···2754···54
size11113311331···133331···133333···33···3

66 irreducible representations

dim111111111133
type++
imageC1C2C3C3C6C6C9C9C18C18C27⋊C3C2×C27⋊C3
kernelC2×C27⋊C3C27⋊C3C54C3×C18C27C3×C9C18C3×C6C9C32C2C1
# reps11626212612666

Matrix representation of C2×C27⋊C3 in GL3(𝔽109) generated by

10800
01080
00108
,
010
0063
2700
,
100
0630
0045
G:=sub<GL(3,GF(109))| [108,0,0,0,108,0,0,0,108],[0,0,27,1,0,0,0,63,0],[1,0,0,0,63,0,0,0,45] >;

C2×C27⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{27}\rtimes C_3
% in TeX

G:=Group("C2xC27:C3");
// GroupNames label

G:=SmallGroup(162,27);
// by ID

G=gap.SmallGroup(162,27);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,96,457,78]);
// Polycyclic

G:=Group<a,b,c|a^2=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

Export

Subgroup lattice of C2×C27⋊C3 in TeX

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