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## G = C2×He3.2C6order 324 = 22·34

### Direct product of C2 and He3.2C6

Aliases: C2×He3.2C6, (C3×C9)⋊7D6, (C3×C18)⋊3S3, He3⋊C22C6, He3.2(C2×C6), (C2×He3).5C6, C32.3(S3×C6), C6.17(C32⋊C6), He3⋊C32C22, (C3×C6).6(C3×S3), C3.8(C2×C32⋊C6), (C2×He3⋊C2)⋊2C3, (C2×He3⋊C3)⋊1C2, SmallGroup(324,72)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×He3.2C6
 Chief series C1 — C3 — C32 — He3 — He3⋊C3 — He3.2C6 — C2×He3.2C6
 Lower central He3 — C2×He3.2C6
 Upper central C1 — C6

Generators and relations for C2×He3.2C6
G = < a,b,c,d,e | a2=b3=c3=d3=1, e6=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1c, cd=dc, ce=ec, ede-1=b-1d-1 >

Subgroups: 268 in 56 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, C18, C3×S3, C3×C6, C3×C6, C3×C9, He3, He3, C2×C18, S3×C6, S3×C9, He3⋊C2, C3×C18, C2×He3, C2×He3, He3⋊C3, S3×C18, C2×He3⋊C2, He3.2C6, C2×He3⋊C3, C2×He3.2C6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, He3.2C6, C2×He3.2C6

Smallest permutation representation of C2×He3.2C6
On 54 points
Generators in S54
(1 18)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)
(1 27 30)(2 19 22)(3 29 32)(4 21 24)(5 31 34)(6 23 26)(7 33 36)(8 25 28)(9 35 20)(10 48 51)(11 40 43)(12 50 53)(13 42 45)(14 52 37)(15 44 47)(16 54 39)(17 46 49)(18 38 41)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(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 30 21)(3 29 20)(4 24 33)(6 23 32)(7 36 27)(9 35 26)(11 40 49)(12 53 44)(14 52 43)(15 47 38)(17 46 37)(18 41 50)(19 25 31)(22 34 28)(39 51 45)(42 48 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)| (1,18)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47), (1,27,30)(2,19,22)(3,29,32)(4,21,24)(5,31,34)(6,23,26)(7,33,36)(8,25,28)(9,35,20)(10,48,51)(11,40,43)(12,50,53)(13,42,45)(14,52,37)(15,44,47)(16,54,39)(17,46,49)(18,38,41), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(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), (1,30,21)(3,29,20)(4,24,33)(6,23,32)(7,36,27)(9,35,26)(11,40,49)(12,53,44)(14,52,43)(15,47,38)(17,46,37)(18,41,50)(19,25,31)(22,34,28)(39,51,45)(42,48,54), (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( (1,18)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47), (1,27,30)(2,19,22)(3,29,32)(4,21,24)(5,31,34)(6,23,26)(7,33,36)(8,25,28)(9,35,20)(10,48,51)(11,40,43)(12,50,53)(13,42,45)(14,52,37)(15,44,47)(16,54,39)(17,46,49)(18,38,41), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(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), (1,30,21)(3,29,20)(4,24,33)(6,23,32)(7,36,27)(9,35,26)(11,40,49)(12,53,44)(14,52,43)(15,47,38)(17,46,37)(18,41,50)(19,25,31)(22,34,28)(39,51,45)(42,48,54), (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([[(1,18),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47)], [(1,27,30),(2,19,22),(3,29,32),(4,21,24),(5,31,34),(6,23,26),(7,33,36),(8,25,28),(9,35,20),(10,48,51),(11,40,43),(12,50,53),(13,42,45),(14,52,37),(15,44,47),(16,54,39),(17,46,49),(18,38,41)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(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)], [(1,30,21),(3,29,20),(4,24,33),(6,23,32),(7,36,27),(9,35,26),(11,40,49),(12,53,44),(14,52,43),(15,47,38),(17,46,37),(18,41,50),(19,25,31),(22,34,28),(39,51,45),(42,48,54)], [(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)]])

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 He3.2C6 C2×He3.2C6 C32⋊C6 C2×C32⋊C6 kernel C2×He3.2C6 He3.2C6 C2×He3⋊C3 C2×He3⋊C2 He3⋊C2 C2×He3 C3×C18 C3×C9 C3×C6 C32 C2 C1 C6 C3 # reps 1 2 1 2 4 2 1 1 2 2 12 12 1 1

Matrix representation of C2×He3.2C6 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 7 0 0 7 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 18 1 0 0 0 18 0 0 0 0 0 0 0 0 1 0 0 11 0 0 0 0 0 7 0
,
 0 11 0 0 0 11 0 0 0 0 0 0 15 15 13 0 0 13 10 13 0 0 15 10 10

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,7,0,0,0,0,0,7,0],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[18,18,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,7,0,0,1,0,0],[0,11,0,0,0,11,0,0,0,0,0,0,15,13,15,0,0,15,10,10,0,0,13,13,10] >;

C2×He3.2C6 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._2C_6
% in TeX

G:=Group("C2xHe3.2C6");
// GroupNames label

G:=SmallGroup(324,72);
// by ID

G=gap.SmallGroup(324,72);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,579,303,5404,382]);
// Polycyclic

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

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