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## G = C2×He3.4C6order 324 = 22·34

### Direct product of C2 and He3.4C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×He3.4C6
 Chief series C1 — C3 — C32 — He3 — C9○He3 — He3.4C6 — C2×He3.4C6
 Lower central He3 — C2×He3.4C6
 Upper central C1 — C18

Generators and relations for C2×He3.4C6
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-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 277 in 101 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C9, C32, D6, C2×C6, C18, C18, C3×S3, C3×C6, C3×C9, He3, 3- 1+2, C2×C18, S3×C6, S3×C9, He3⋊C2, C3×C18, C2×He3, C2×3- 1+2, C9○He3, S3×C18, C2×He3⋊C2, He3.4C6, C2×C9○He3, C2×He3.4C6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C6×C3⋊S3, He3.4C6, C2×He3.4C6

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

60 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 9G ··· 9N 18A ··· 18F 18G ··· 18N 18O ··· 18Z order 1 2 2 2 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 9 9 1 1 6 6 6 6 1 1 6 6 6 6 9 9 9 9 1 ··· 1 6 ··· 6 1 ··· 1 6 ··· 6 9 ··· 9

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 He3.4C6 C2×He3.4C6 kernel C2×He3.4C6 He3.4C6 C2×C9○He3 C2×He3⋊C2 He3⋊C2 C2×He3 C3×C18 C3×C9 C3×C6 C32 C2 C1 # reps 1 2 1 2 4 2 4 4 8 8 12 12

Matrix representation of C2×He3.4C6 in GL3(𝔽19) generated by

 18 0 0 0 18 0 0 0 18
,
 1 10 0 0 18 1 0 18 0
,
 7 0 0 0 7 0 0 0 7
,
 1 0 0 1 11 0 12 0 7
,
 3 0 0 0 0 3 0 3 0
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[1,0,0,10,18,18,0,1,0],[7,0,0,0,7,0,0,0,7],[1,1,12,0,11,0,0,0,7],[3,0,0,0,0,3,0,3,0] >;

C2×He3.4C6 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._4C_6
% in TeX

G:=Group("C2xHe3.4C6");
// GroupNames label

G:=SmallGroup(324,148);
// by ID

G=gap.SmallGroup(324,148);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,579,2164,382]);
// 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*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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