direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×He3⋊C3, He3⋊3C6, C6.4He3, (C3×C9)⋊10C6, (C3×C18)⋊3C3, (C2×He3)⋊2C3, C3.4(C2×He3), C32.3(C3×C6), (C3×C6).3C32, SmallGroup(162,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3⋊C3
G = < a,b,c,d,e | a2=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ce=ec, ede-1=bcd >
Smallest permutation representation of C2×He3⋊C3
►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)
(2 15 11)(3 12 13)(4 9 7)(5 6 54)(8 52 53)(16 18 20)(17 23 22)(19 21 24)(25 26 31)(27 28 29)(30 32 33)(35 42 38)(36 39 40)(43 45 47)(44 50 49)(46 48 51)
(1 28 20)(2 26 22)(3 31 16)(4 50 40)(5 47 34)(6 45 38)(7 48 42)(8 43 36)(9 51 37)(10 32 24)(11 29 18)(12 27 19)(13 30 23)(14 25 17)(15 33 21)(35 53 49)(39 54 46)(41 52 44)
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), (2,15,11)(3,12,13)(4,9,7)(5,6,54)(8,52,53)(16,18,20)(17,23,22)(19,21,24)(25,26,31)(27,28,29)(30,32,33)(35,42,38)(36,39,40)(43,45,47)(44,50,49)(46,48,51), (1,28,20)(2,26,22)(3,31,16)(4,50,40)(5,47,34)(6,45,38)(7,48,42)(8,43,36)(9,51,37)(10,32,24)(11,29,18)(12,27,19)(13,30,23)(14,25,17)(15,33,21)(35,53,49)(39,54,46)(41,52,44)>;
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), (2,15,11)(3,12,13)(4,9,7)(5,6,54)(8,52,53)(16,18,20)(17,23,22)(19,21,24)(25,26,31)(27,28,29)(30,32,33)(35,42,38)(36,39,40)(43,45,47)(44,50,49)(46,48,51), (1,28,20)(2,26,22)(3,31,16)(4,50,40)(5,47,34)(6,45,38)(7,48,42)(8,43,36)(9,51,37)(10,32,24)(11,29,18)(12,27,19)(13,30,23)(14,25,17)(15,33,21)(35,53,49)(39,54,46)(41,52,44) );
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)], [(2,15,11),(3,12,13),(4,9,7),(5,6,54),(8,52,53),(16,18,20),(17,23,22),(19,21,24),(25,26,31),(27,28,29),(30,32,33),(35,42,38),(36,39,40),(43,45,47),(44,50,49),(46,48,51)], [(1,28,20),(2,26,22),(3,31,16),(4,50,40),(5,47,34),(6,45,38),(7,48,42),(8,43,36),(9,51,37),(10,32,24),(11,29,18),(12,27,19),(13,30,23),(14,25,17),(15,33,21),(35,53,49),(39,54,46),(41,52,44)]])
C2×He3⋊C3 is a maximal subgroup of
He3.2C12 He3.2Dic3 He3⋊Dic3
34 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 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 | 3 | ··· | 3 | 3 | ··· | 3 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | He3 | C2×He3 | He3⋊C3 | C2×He3⋊C3 |
| kernel | C2×He3⋊C3 | He3⋊C3 | C3×C18 | C2×He3 | C3×C9 | He3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 6 | 6 |
Matrix representation of C2×He3⋊C3 ►in GL3(𝔽19) generated by
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 11 | 10 | 0 |
| 0 | 8 | 1 |
| 0 | 12 | 0 |
| 11 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 18 | 7 | 0 |
| 7 | 0 | 11 |
| 4 | 13 | 13 |
| 15 | 0 | 10 |
| 9 | 9 | 15 |
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[11,0,0,10,8,12,0,1,0],[11,0,0,0,11,0,0,0,11],[1,18,7,0,7,0,0,0,11],[4,15,9,13,0,9,13,10,15] >;
C2×He3⋊C3 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes C_3 % in TeX
G:=Group("C2xHe3:C3"); // GroupNames label
G:=SmallGroup(162,30);
// by ID
G=gap.SmallGroup(162,30);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,187,282,728]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b*c^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
// generators/relations
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