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## G = C2×C32⋊He3order 486 = 2·35

### Direct product of C2 and C32⋊He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C2×C32⋊He3, C3411C6, (C3×C6)⋊He3, (C6×He3)⋊2C3, C334(C3×C6), (C33×C6)⋊1C3, C6.3(C3×He3), C3.3(C6×He3), (C3×He3)⋊14C6, C322(C2×He3), (C3×C6).20C33, (C32×C6)⋊1C32, C32.24(C32×C6), SmallGroup(486,196)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32⋊He3
 Chief series C1 — C3 — C32 — C33 — C34 — C32⋊He3 — C2×C32⋊He3
 Lower central C1 — C32 — C2×C32⋊He3
 Upper central C1 — C3×C6 — C2×C32⋊He3

Generators and relations for C2×C32⋊He3
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 1062 in 378 conjugacy classes, 90 normal (8 characteristic)
C1, C2, C3, C3, C6, C6, C32, C32, C32, C3×C6, C3×C6, C3×C6, He3, C33, C33, C2×He3, C32×C6, C32×C6, C3×He3, C34, C6×He3, C33×C6, C32⋊He3, C2×C32⋊He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C32⋊He3, C2×C32⋊He3

Smallest permutation representation of C2×C32⋊He3
On 54 points
Generators in S54
(1 38)(2 39)(3 37)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(10 35)(11 36)(12 34)(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 15 11)(2 13 12)(3 14 10)(4 53 8)(5 54 9)(6 52 7)(16 24 20)(17 22 21)(18 23 19)(25 33 29)(26 31 30)(27 32 28)(34 39 40)(35 37 41)(36 38 42)(43 51 47)(44 49 48)(45 50 46)
(1 3 2)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 24 20)(17 22 21)(18 23 19)(25 26 27)(28 29 30)(31 32 33)(34 36 35)(37 39 38)(40 42 41)(43 51 47)(44 49 48)(45 50 46)(52 53 54)
(1 13 10)(2 14 11)(3 15 12)(4 9 52)(5 7 53)(6 8 54)(16 17 18)(19 20 21)(22 23 24)(25 30 32)(26 28 33)(27 29 31)(34 37 42)(35 38 40)(36 39 41)(43 44 45)(46 47 48)(49 50 51)
(1 31 17)(2 28 19)(3 25 24)(4 49 42)(5 46 39)(6 43 35)(7 47 41)(8 44 38)(9 50 34)(10 29 16)(11 26 21)(12 32 23)(13 27 18)(14 33 20)(15 30 22)(36 53 48)(37 52 51)(40 54 45)

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

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

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

102 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AF 3AG ··· 3AX 6A ··· 6H 6I ··· 6AF 6AG ··· 6AX order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 ··· 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 kernel C2×C32⋊He3 C32⋊He3 C6×He3 C33×C6 C3×He3 C34 C3×C6 C32 # reps 1 1 24 2 24 2 24 24

Matrix representation of C2×C32⋊He3 in GL6(𝔽7)

 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 3 4 2
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 4 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 5 3 6 0 0 0 0 0 4

G:=sub<GL(6,GF(7))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,3,0,0,0,0,1,4,0,0,0,0,0,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,1,3,0,0,0,0,0,6,4] >;

C2×C32⋊He3 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes {\rm He}_3
% in TeX

G:=Group("C2xC3^2:He3");
// GroupNames label

G:=SmallGroup(486,196);
// by ID

G=gap.SmallGroup(486,196);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,2169]);
// Polycyclic

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

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