p-group, metabelian, nilpotent (class 3), monomial
Aliases: C32.19He3, C33.19C32, C32.53- 1+2, (C3×C9)⋊2C9, C32⋊C9.5C3, C32.8(C3×C9), (C32×C9).3C3, C3.4(C32⋊C9), C3.1(He3.C3), SmallGroup(243,14)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32.19He3
G = < a,b,c,d,e | a3=b3=d3=1, c3=b, e3=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=bcd-1, ede-1=b-1d >
Smallest permutation representation of C32.19He3
►On 81 points
Generators in S81
(1 41 34)(2 42 35)(3 43 36)(4 44 28)(5 45 29)(6 37 30)(7 38 31)(8 39 32)(9 40 33)(10 81 26)(11 73 27)(12 74 19)(13 75 20)(14 76 21)(15 77 22)(16 78 23)(17 79 24)(18 80 25)(46 60 68)(47 61 69)(48 62 70)(49 63 71)(50 55 72)(51 56 64)(52 57 65)(53 58 66)(54 59 67)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)
(1 41 34)(2 42 35)(3 43 36)(4 44 28)(5 45 29)(6 37 30)(7 38 31)(8 39 32)(9 40 33)(10 78 20)(11 79 21)(12 80 22)(13 81 23)(14 73 24)(15 74 25)(16 75 26)(17 76 27)(18 77 19)(46 63 65)(47 55 66)(48 56 67)(49 57 68)(50 58 69)(51 59 70)(52 60 71)(53 61 72)(54 62 64)
(1 78 51 41 23 56 34 16 64)(2 21 65 42 14 52 35 76 57)(3 12 58 43 74 66 36 19 53)(4 81 54 44 26 59 28 10 67)(5 24 68 45 17 46 29 79 60)(6 15 61 37 77 69 30 22 47)(7 75 48 38 20 62 31 13 70)(8 27 71 39 11 49 32 73 63)(9 18 55 40 80 72 33 25 50)
G:=sub<Sym(81)| (1,41,34)(2,42,35)(3,43,36)(4,44,28)(5,45,29)(6,37,30)(7,38,31)(8,39,32)(9,40,33)(10,81,26)(11,73,27)(12,74,19)(13,75,20)(14,76,21)(15,77,22)(16,78,23)(17,79,24)(18,80,25)(46,60,68)(47,61,69)(48,62,70)(49,63,71)(50,55,72)(51,56,64)(52,57,65)(53,58,66)(54,59,67), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,41,34)(2,42,35)(3,43,36)(4,44,28)(5,45,29)(6,37,30)(7,38,31)(8,39,32)(9,40,33)(10,78,20)(11,79,21)(12,80,22)(13,81,23)(14,73,24)(15,74,25)(16,75,26)(17,76,27)(18,77,19)(46,63,65)(47,55,66)(48,56,67)(49,57,68)(50,58,69)(51,59,70)(52,60,71)(53,61,72)(54,62,64), (1,78,51,41,23,56,34,16,64)(2,21,65,42,14,52,35,76,57)(3,12,58,43,74,66,36,19,53)(4,81,54,44,26,59,28,10,67)(5,24,68,45,17,46,29,79,60)(6,15,61,37,77,69,30,22,47)(7,75,48,38,20,62,31,13,70)(8,27,71,39,11,49,32,73,63)(9,18,55,40,80,72,33,25,50)>;
G:=Group( (1,41,34)(2,42,35)(3,43,36)(4,44,28)(5,45,29)(6,37,30)(7,38,31)(8,39,32)(9,40,33)(10,81,26)(11,73,27)(12,74,19)(13,75,20)(14,76,21)(15,77,22)(16,78,23)(17,79,24)(18,80,25)(46,60,68)(47,61,69)(48,62,70)(49,63,71)(50,55,72)(51,56,64)(52,57,65)(53,58,66)(54,59,67), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,41,34)(2,42,35)(3,43,36)(4,44,28)(5,45,29)(6,37,30)(7,38,31)(8,39,32)(9,40,33)(10,78,20)(11,79,21)(12,80,22)(13,81,23)(14,73,24)(15,74,25)(16,75,26)(17,76,27)(18,77,19)(46,63,65)(47,55,66)(48,56,67)(49,57,68)(50,58,69)(51,59,70)(52,60,71)(53,61,72)(54,62,64), (1,78,51,41,23,56,34,16,64)(2,21,65,42,14,52,35,76,57)(3,12,58,43,74,66,36,19,53)(4,81,54,44,26,59,28,10,67)(5,24,68,45,17,46,29,79,60)(6,15,61,37,77,69,30,22,47)(7,75,48,38,20,62,31,13,70)(8,27,71,39,11,49,32,73,63)(9,18,55,40,80,72,33,25,50) );
G=PermutationGroup([[(1,41,34),(2,42,35),(3,43,36),(4,44,28),(5,45,29),(6,37,30),(7,38,31),(8,39,32),(9,40,33),(10,81,26),(11,73,27),(12,74,19),(13,75,20),(14,76,21),(15,77,22),(16,78,23),(17,79,24),(18,80,25),(46,60,68),(47,61,69),(48,62,70),(49,63,71),(50,55,72),(51,56,64),(52,57,65),(53,58,66),(54,59,67)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81)], [(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)], [(1,41,34),(2,42,35),(3,43,36),(4,44,28),(5,45,29),(6,37,30),(7,38,31),(8,39,32),(9,40,33),(10,78,20),(11,79,21),(12,80,22),(13,81,23),(14,73,24),(15,74,25),(16,75,26),(17,76,27),(18,77,19),(46,63,65),(47,55,66),(48,56,67),(49,57,68),(50,58,69),(51,59,70),(52,60,71),(53,61,72),(54,62,64)], [(1,78,51,41,23,56,34,16,64),(2,21,65,42,14,52,35,76,57),(3,12,58,43,74,66,36,19,53),(4,81,54,44,26,59,28,10,67),(5,24,68,45,17,46,29,79,60),(6,15,61,37,77,69,30,22,47),(7,75,48,38,20,62,31,13,70),(8,27,71,39,11,49,32,73,63),(9,18,55,40,80,72,33,25,50)]])
C32.19He3 is a maximal subgroup of
(C3×C9)⋊C18 (C3×C9)⋊D9 (C3×C9)⋊5D9
51 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 9A | ··· | 9R | 9S | ··· | 9AJ |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C9 | He3 | 3- 1+2 | He3.C3 |
| kernel | C32.19He3 | C32⋊C9 | C32×C9 | C3×C9 | C32 | C32 | C3 |
| # reps | 1 | 6 | 2 | 18 | 2 | 4 | 18 |
Matrix representation of C32.19He3 ►in GL4(𝔽19) generated by
| 11 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 11 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 11 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(19))| [11,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[11,0,0,0,0,9,0,0,0,0,9,0,0,0,0,6],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,0,11],[5,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C32.19He3 in GAP, Magma, Sage, TeX
C_3^2._{19}{\rm He}_3 % in TeX
G:=Group("C3^2.19He3"); // GroupNames label
G:=SmallGroup(243,14);
// by ID
G=gap.SmallGroup(243,14);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,276,1352]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=d^3=1,c^3=b,e^3=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b*c*d^-1,e*d*e^-1=b^-1*d>;
// generators/relations
Export