direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×He3⋊C32, He3⋊4(C3×C6), (C6×He3)⋊6C3, (C3×He3)⋊18C6, (C3×C18)⋊3C32, (C3×C6).11He3, C3.14(C6×He3), C6.14(C3×He3), (C3×C6).8C33, He3⋊C3⋊7C6, C33.16(C3×C6), (C2×He3)⋊3C32, C32.11(C2×He3), C32.8(C32×C6), (C32×C6).15C32, (C6×3- 1+2)⋊9C3, (C3×3- 1+2)⋊16C6, (C3×C9)⋊7(C3×C6), (C2×He3⋊C3)⋊3C3, SmallGroup(486,217)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3⋊C32
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, dbd-1=ebe-1=bc-1, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=bd, fdf-1=c-1d, ef=fe >
Subgroups: 576 in 160 conjugacy classes, 66 normal (12 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, He3, He3, 3- 1+2, C33, C33, C3×C18, C2×He3, C2×He3, C2×3- 1+2, C32×C6, C32×C6, He3⋊C3, C3×He3, C3×3- 1+2, C2×He3⋊C3, C6×He3, C6×3- 1+2, He3⋊C32, C2×He3⋊C32
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, He3⋊C32, C2×He3⋊C32
►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)
(1 32 20)(2 29 22)(3 27 16)(4 46 39)(5 44 41)(6 49 35)(7 45 38)(8 50 40)(9 47 34)(10 25 24)(11 33 18)(12 30 19)(13 31 23)(14 28 17)(15 26 21)(36 54 43)(37 52 51)(42 53 48)
(1 16 31)(2 24 25)(3 21 29)(4 41 50)(5 38 44)(6 36 48)(7 39 49)(8 34 43)(9 42 47)(10 19 27)(11 17 28)(12 22 33)(13 18 26)(14 23 30)(15 20 32)(35 51 52)(37 46 54)(40 45 53)
(1 2 3)(4 9 53)(5 7 54)(6 8 52)(10 11 12)(13 14 15)(16 24 21)(17 22 19)(18 23 20)(25 29 31)(26 30 32)(27 28 33)(34 35 36)(37 38 39)(40 41 42)(43 51 48)(44 49 46)(45 50 47)
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), (1,32,20)(2,29,22)(3,27,16)(4,46,39)(5,44,41)(6,49,35)(7,45,38)(8,50,40)(9,47,34)(10,25,24)(11,33,18)(12,30,19)(13,31,23)(14,28,17)(15,26,21)(36,54,43)(37,52,51)(42,53,48), (1,16,31)(2,24,25)(3,21,29)(4,41,50)(5,38,44)(6,36,48)(7,39,49)(8,34,43)(9,42,47)(10,19,27)(11,17,28)(12,22,33)(13,18,26)(14,23,30)(15,20,32)(35,51,52)(37,46,54)(40,45,53), (1,2,3)(4,9,53)(5,7,54)(6,8,52)(10,11,12)(13,14,15)(16,24,21)(17,22,19)(18,23,20)(25,29,31)(26,30,32)(27,28,33)(34,35,36)(37,38,39)(40,41,42)(43,51,48)(44,49,46)(45,50,47)>;
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), (1,32,20)(2,29,22)(3,27,16)(4,46,39)(5,44,41)(6,49,35)(7,45,38)(8,50,40)(9,47,34)(10,25,24)(11,33,18)(12,30,19)(13,31,23)(14,28,17)(15,26,21)(36,54,43)(37,52,51)(42,53,48), (1,16,31)(2,24,25)(3,21,29)(4,41,50)(5,38,44)(6,36,48)(7,39,49)(8,34,43)(9,42,47)(10,19,27)(11,17,28)(12,22,33)(13,18,26)(14,23,30)(15,20,32)(35,51,52)(37,46,54)(40,45,53), (1,2,3)(4,9,53)(5,7,54)(6,8,52)(10,11,12)(13,14,15)(16,24,21)(17,22,19)(18,23,20)(25,29,31)(26,30,32)(27,28,33)(34,35,36)(37,38,39)(40,41,42)(43,51,48)(44,49,46)(45,50,47) );
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)], [(1,32,20),(2,29,22),(3,27,16),(4,46,39),(5,44,41),(6,49,35),(7,45,38),(8,50,40),(9,47,34),(10,25,24),(11,33,18),(12,30,19),(13,31,23),(14,28,17),(15,26,21),(36,54,43),(37,52,51),(42,53,48)], [(1,16,31),(2,24,25),(3,21,29),(4,41,50),(5,38,44),(6,36,48),(7,39,49),(8,34,43),(9,42,47),(10,19,27),(11,17,28),(12,22,33),(13,18,26),(14,23,30),(15,20,32),(35,51,52),(37,46,54),(40,45,53)], [(1,2,3),(4,9,53),(5,7,54),(6,8,52),(10,11,12),(13,14,15),(16,24,21),(17,22,19),(18,23,20),(25,29,31),(26,30,32),(27,28,33),(34,35,36),(37,38,39),(40,41,42),(43,51,48),(44,49,46),(45,50,47)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 3K | ··· | 3AB | 6A | 6B | 6C | ··· | 6J | 6K | ··· | 6AB | 9A | ··· | 9F | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 1 | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 9 | 9 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | He3 | C2×He3 | He3⋊C32 | C2×He3⋊C32 |
| kernel | C2×He3⋊C32 | He3⋊C32 | C2×He3⋊C3 | C6×He3 | C6×3- 1+2 | He3⋊C3 | C3×He3 | C3×3- 1+2 | C3×C6 | C32 | C2 | C1 |
| # reps | 1 | 1 | 18 | 6 | 2 | 18 | 6 | 2 | 6 | 6 | 2 | 2 |
Matrix representation of C2×He3⋊C32 ►in GL9(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 7 |
| 11 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 12 | 0 | 0 | 1 | 0 | 12 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 11 | 12 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 8 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 8 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 18 | 0 | 0 | 0 |
| 7 | 0 | 11 | 0 | 0 | 18 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 7 |
| 7 | 0 | 0 | 0 | 0 | 0 | 1 | 8 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 18 | 0 |
| 12 | 0 | 7 | 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 12 | 0 |
| 8 | 0 | 0 | 7 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 12 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 12 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 7 | 12 | 0 | 0 | 0 | 7 | 0 | 0 | 0 |
| 7 | 12 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 12 | 8 | 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 12 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 12 | 8 | 0 | 0 | 0 | 0 | 11 | 0 | 0 |
G:=sub<GL(9,GF(19))| [18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,0,0,6,18,18,12,12,12,8,8,8,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[11,12,11,8,1,0,0,18,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,4,12,12,8,8,8,18,18,18,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0],[7,18,7,11,0,12,0,8,1,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,1,0,0,0,0,0,0,9,8,8,18,18,18,12,12,12,0,7,0,0,0,0,0,0,0],[1,0,0,7,7,7,12,12,12,6,18,18,12,12,12,8,8,8,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0] >;
C2×He3⋊C32 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes C_3^2 % in TeX
G:=Group("C2xHe3:C3^2"); // GroupNames label
G:=SmallGroup(486,217);
// by ID
G=gap.SmallGroup(486,217);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,986,735,3250]);
// 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,d*b*d^-1=e*b*e^-1=b*c^-1,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=b*d,f*d*f^-1=c^-1*d,e*f=f*e>;
// generators/relations