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G = C2×C9.4He3order 486 = 2·35

Direct product of C2 and C9.4He3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C2×C9.4He3, C18.3He3, C33.4C18, C18.43- 1+2, C27⋊C31C6, (C3×C18).3C9, (C3×C9).5C18, C9.3(C2×He3), (C32×C6).2C9, C6.6(C32⋊C9), (C32×C9).22C6, (C32×C18).10C3, C32.10(C3×C18), (C3×C18).23C32, C9.4(C2×3- 1+2), (C2×C27⋊C3)⋊1C3, (C3×C6).10(C3×C9), (C3×C9).32(C3×C6), C3.6(C2×C32⋊C9), SmallGroup(486,76)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C9.4He3
C1C3C9C3×C9C32×C9C9.4He3 — C2×C9.4He3
C1C3C32 — C2×C9.4He3
C1C18C3×C18 — C2×C9.4He3

Generators and relations for C2×C9.4He3
 G = < a,b,c,d,e | a2=b9=c3=d3=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3cd-1, ede-1=b6d >

Subgroups: 126 in 62 conjugacy classes, 30 normal (20 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C9, C32, C32, C18, C18, C18, C3×C6, C3×C6, C27, C3×C9, C3×C9, C33, C54, C3×C18, C3×C18, C32×C6, C27⋊C3, C32×C9, C2×C27⋊C3, C32×C18, C9.4He3, C2×C9.4He3
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, 3- 1+2, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, C2×C32⋊C9, C9.4He3, C2×C9.4He3

Smallest permutation representation of C2×C9.4He3
On 54 points
Generators in S54
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)
(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)
(1 19 10)(2 11 20)(3 12 21)(4 22 13)(5 14 23)(6 15 24)(7 25 16)(8 17 26)(9 18 27)(28 37 46)(29 47 38)(30 39 48)(31 40 49)(32 50 41)(33 42 51)(34 43 52)(35 53 44)(36 45 54)
(1 19 10)(2 11 20)(4 22 13)(5 14 23)(7 25 16)(8 17 26)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 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)

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

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

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

102 conjugacy classes

class 1  2 3A3B3C···3J6A6B6C···6J9A···9F9G···9V18A···18F18G···18V27A···27R54A···54R
order12333···3666···69···99···918···1818···1827···2754···54
size11113···3113···31···13···31···13···39···99···9

102 irreducible representations

dim1111111111333333
type++
imageC1C2C3C3C6C6C9C9C18C18He33- 1+2C2×He3C2×3- 1+2C9.4He3C2×C9.4He3
kernelC2×C9.4He3C9.4He3C2×C27⋊C3C32×C18C27⋊C3C32×C9C3×C18C32×C6C3×C9C33C18C18C9C9C2C1
# reps11626212612624241818

Matrix representation of C2×C9.4He3 in GL4(𝔽109) generated by

108000
0100
0010
0001
,
1000
010500
001050
000105
,
63000
0634526
00450
00045
,
1000
0634626
0010
00045
,
1000
071450
0001
07934102
G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,105,0,0,0,0,105,0,0,0,0,105],[63,0,0,0,0,63,0,0,0,45,45,0,0,26,0,45],[1,0,0,0,0,63,0,0,0,46,1,0,0,26,0,45],[1,0,0,0,0,7,0,79,0,14,0,34,0,50,1,102] >;

C2×C9.4He3 in GAP, Magma, Sage, TeX

C_2\times C_9._4{\rm He}_3
% in TeX

G:=Group("C2xC9.4He3");
// GroupNames label

G:=SmallGroup(486,76);
// by ID

G=gap.SmallGroup(486,76);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,2169,118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=c^3=d^3=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,c*d=d*c,e*c*e^-1=b^3*c*d^-1,e*d*e^-1=b^6*d>;
// generators/relations

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