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G = C2×He3⋊C3order 162 = 2·34

Direct product of C2 and He3⋊C3

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

Aliases: C2×He3⋊C3, He33C6, C6.4He3, (C3×C9)⋊10C6, (C3×C18)⋊3C3, (C2×He3)⋊2C3, C3.4(C2×He3), C32.3(C3×C6), (C3×C6).3C32, SmallGroup(162,30)

Series: Derived Chief Lower central Upper central

C1C32 — C2×He3⋊C3
C1C3C32C3×C9He3⋊C3 — C2×He3⋊C3
C1C3C32 — C2×He3⋊C3
C1C6C3×C6 — C2×He3⋊C3

Generators and relations for C2×He3⋊C3
 G = < a,b,c,d,e | a2=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ce=ec, ede-1=bcd >

3C3
9C3
9C3
9C3
3C6
9C6
9C6
9C6
3C32
3C32
3C32
3C9
3C3×C6
3C3×C6
3C3×C6
3C18

Smallest permutation representation of C2×He3⋊C3
On 54 points
Generators in S54
(1 34)(2 35)(3 36)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(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)
(1 14 10)(2 15 11)(3 13 12)(4 54 8)(5 52 9)(6 53 7)(16 23 19)(17 24 20)(18 22 21)(25 32 28)(26 33 29)(27 31 30)(34 41 37)(35 42 38)(36 40 39)(43 50 46)(44 51 47)(45 49 48)
(2 15 11)(3 12 13)(4 9 7)(5 6 54)(8 52 53)(16 18 20)(17 23 22)(19 21 24)(25 26 31)(27 28 29)(30 32 33)(35 42 38)(36 39 40)(43 45 47)(44 50 49)(46 48 51)
(1 28 20)(2 26 22)(3 31 16)(4 50 40)(5 47 34)(6 45 38)(7 48 42)(8 43 36)(9 51 37)(10 32 24)(11 29 18)(12 27 19)(13 30 23)(14 25 17)(15 33 21)(35 53 49)(39 54 46)(41 52 44)

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

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

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

C2×He3⋊C3 is a maximal subgroup of   He3.2C12  He3.2Dic3  He3⋊Dic3

34 conjugacy classes

class 1  2 3A3B3C3D3E···3J6A6B6C6D6E···6J9A···9F18A···18F
order1233333···366666···69···918···18
size1111339···911339···93···33···3

34 irreducible representations

dim1111113333
type++
imageC1C2C3C3C6C6He3C2×He3He3⋊C3C2×He3⋊C3
kernelC2×He3⋊C3He3⋊C3C3×C18C2×He3C3×C9He3C6C3C2C1
# reps1126262266

Matrix representation of C2×He3⋊C3 in GL3(𝔽19) generated by

1800
0180
0018
,
11100
081
0120
,
1100
0110
0011
,
100
1870
7011
,
41313
15010
9915
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[11,0,0,10,8,12,0,1,0],[11,0,0,0,11,0,0,0,11],[1,18,7,0,7,0,0,0,11],[4,15,9,13,0,9,13,10,15] >;

C2×He3⋊C3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes C_3
% in TeX

G:=Group("C2xHe3:C3");
// GroupNames label

G:=SmallGroup(162,30);
// by ID

G=gap.SmallGroup(162,30);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,187,282,728]);
// Polycyclic

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

Export

Subgroup lattice of C2×He3⋊C3 in TeX

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