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## G = (C3×He3).C6order 486 = 2·35

### 2nd non-split extension by C3×He3 of C6 acting faithfully

Aliases: C32⋊C92S3, (C3×He3).2C6, C33.6(C3×S3), He35S3.2C3, C3.5(C33⋊C6), C32.27He31C2, C32.30(C32⋊C6), C3.2(He3.2C6), SmallGroup(486,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — (C3×He3).C6
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C32.27He3 — (C3×He3).C6
 Lower central C3×He3 — (C3×He3).C6
 Upper central C1 — C3

Generators and relations for (C3×He3).C6
G = < a,b,c,d,e | a3=b3=c3=d3=1, e6=c, ab=ba, ac=ca, ad=da, eae-1=a-1c, bc=cb, dbd-1=bc-1, ebe-1=a-1b-1, cd=dc, ce=ec, ede-1=a-1b-1c-1d-1 >

Subgroups: 524 in 54 conjugacy classes, 10 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, He3, C33, C33, S3×C9, He3⋊C2, C3×C3⋊S3, C32⋊C9, C32⋊C9, C3×He3, C32⋊C18, He35S3, C32.27He3, (C3×He3).C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C32⋊C6, C33⋊C6, He3.2C6, (C3×He3).C6

Smallest permutation representation of (C3×He3).C6
On 54 points
Generators in S54
(1 7 13)(3 9 15)(5 11 17)(20 26 32)(22 28 34)(24 30 36)(38 44 50)(40 46 52)(42 48 54)
(1 7 13)(2 8 14)(3 15 9)(6 18 12)(21 33 27)(22 28 34)(23 29 35)(24 36 30)(39 51 45)(40 46 52)(41 47 53)(42 54 48)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)
(1 50 24)(2 25 45)(3 52 32)(4 27 53)(5 48 28)(6 35 37)(7 38 30)(8 31 51)(9 40 20)(10 33 41)(11 54 34)(12 23 43)(13 44 36)(14 19 39)(15 46 26)(16 21 47)(17 42 22)(18 29 49)
(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)| (1,7,13)(3,9,15)(5,11,17)(20,26,32)(22,28,34)(24,30,36)(38,44,50)(40,46,52)(42,48,54), (1,7,13)(2,8,14)(3,15,9)(6,18,12)(21,33,27)(22,28,34)(23,29,35)(24,36,30)(39,51,45)(40,46,52)(41,47,53)(42,54,48), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54), (1,50,24)(2,25,45)(3,52,32)(4,27,53)(5,48,28)(6,35,37)(7,38,30)(8,31,51)(9,40,20)(10,33,41)(11,54,34)(12,23,43)(13,44,36)(14,19,39)(15,46,26)(16,21,47)(17,42,22)(18,29,49), (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( (1,7,13)(3,9,15)(5,11,17)(20,26,32)(22,28,34)(24,30,36)(38,44,50)(40,46,52)(42,48,54), (1,7,13)(2,8,14)(3,15,9)(6,18,12)(21,33,27)(22,28,34)(23,29,35)(24,36,30)(39,51,45)(40,46,52)(41,47,53)(42,54,48), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54), (1,50,24)(2,25,45)(3,52,32)(4,27,53)(5,48,28)(6,35,37)(7,38,30)(8,31,51)(9,40,20)(10,33,41)(11,54,34)(12,23,43)(13,44,36)(14,19,39)(15,46,26)(16,21,47)(17,42,22)(18,29,49), (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([[(1,7,13),(3,9,15),(5,11,17),(20,26,32),(22,28,34),(24,30,36),(38,44,50),(40,46,52),(42,48,54)], [(1,7,13),(2,8,14),(3,15,9),(6,18,12),(21,33,27),(22,28,34),(23,29,35),(24,36,30),(39,51,45),(40,46,52),(41,47,53),(42,54,48)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,43,49),(38,44,50),(39,45,51),(40,46,52),(41,47,53),(42,48,54)], [(1,50,24),(2,25,45),(3,52,32),(4,27,53),(5,48,28),(6,35,37),(7,38,30),(8,31,51),(9,40,20),(10,33,41),(11,54,34),(12,23,43),(13,44,36),(14,19,39),(15,46,26),(16,21,47),(17,42,22),(18,29,49)], [(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)]])

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 9A ··· 9F 9G ··· 9L 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 2 2 18 18 18 18 27 27 9 ··· 9 18 ··· 18 27 ··· 27

31 irreducible representations

 dim 1 1 1 1 2 2 3 6 6 6 type + + + + + image C1 C2 C3 C6 S3 C3×S3 He3.2C6 C32⋊C6 C33⋊C6 (C3×He3).C6 kernel (C3×He3).C6 C32.27He3 He3⋊5S3 C3×He3 C32⋊C9 C33 C3 C32 C3 C1 # reps 1 1 2 2 1 2 12 1 3 6

Matrix representation of (C3×He3).C6 in GL6(𝔽19)

 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 0 11 0 0 0 0 0 0 11 0 0 0 11 0 0 0 0 0 0 0 0 0 0 7 0 0 0 7 0 0 0 0 0 0 7 0
,
 0 0 0 0 9 0 0 0 0 0 0 6 0 0 0 9 0 0 0 9 0 0 0 0 0 0 6 0 0 0 9 0 0 0 0 0

G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,11,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,7,0,0],[0,0,0,0,0,9,0,0,0,9,0,0,0,0,0,0,6,0,0,0,9,0,0,0,9,0,0,0,0,0,0,6,0,0,0,0] >;

(C3×He3).C6 in GAP, Magma, Sage, TeX

(C_3\times {\rm He}_3).C_6
% in TeX

G:=Group("(C3xHe3).C6");
// GroupNames label

G:=SmallGroup(486,9);
// by ID

G=gap.SmallGroup(486,9);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,218,224,4755,873,735,3244]);
// Polycyclic

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

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