p-group, metabelian, nilpotent (class 3), monomial
Aliases: C9.6He3, He3.2C9, C9.23- 1+2, 3- 1+2.2C9, C27⋊C3⋊3C3, (C3×C27)⋊3C3, C9○He3.2C3, C32.4(C3×C9), (C3×C9).26C32, C3.10(C32⋊C9), SmallGroup(243,20)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C9.6He3
G = < a,b,c,d | a9=b3=c3=1, d3=a, ab=ba, ac=ca, ad=da, cbc-1=a6b, dbd-1=a6bc-1, dcd-1=a3c >
Smallest permutation representation of C9.6He3
►On 81 points
Generators in S81
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)(55 58 61 64 67 70 73 76 79)(56 59 62 65 68 71 74 77 80)(57 60 63 66 69 72 75 78 81)
(2 77 38)(3 48 78)(5 80 41)(6 51 81)(8 56 44)(9 54 57)(11 59 47)(12 30 60)(14 62 50)(15 33 63)(17 65 53)(18 36 66)(20 68 29)(21 39 69)(23 71 32)(24 42 72)(26 74 35)(27 45 75)(28 46 37)(31 49 40)(34 52 43)(55 64 73)(58 67 76)(61 70 79)
(1 46 58)(2 29 77)(3 39 69)(4 49 61)(5 32 80)(6 42 72)(7 52 64)(8 35 56)(9 45 75)(10 28 67)(11 38 59)(12 48 78)(13 31 70)(14 41 62)(15 51 81)(16 34 73)(17 44 65)(18 54 57)(19 37 76)(20 47 68)(21 30 60)(22 40 79)(23 50 71)(24 33 63)(25 43 55)(26 53 74)(27 36 66)
(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)
G:=sub<Sym(81)| (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81), (2,77,38)(3,48,78)(5,80,41)(6,51,81)(8,56,44)(9,54,57)(11,59,47)(12,30,60)(14,62,50)(15,33,63)(17,65,53)(18,36,66)(20,68,29)(21,39,69)(23,71,32)(24,42,72)(26,74,35)(27,45,75)(28,46,37)(31,49,40)(34,52,43)(55,64,73)(58,67,76)(61,70,79), (1,46,58)(2,29,77)(3,39,69)(4,49,61)(5,32,80)(6,42,72)(7,52,64)(8,35,56)(9,45,75)(10,28,67)(11,38,59)(12,48,78)(13,31,70)(14,41,62)(15,51,81)(16,34,73)(17,44,65)(18,54,57)(19,37,76)(20,47,68)(21,30,60)(22,40,79)(23,50,71)(24,33,63)(25,43,55)(26,53,74)(27,36,66), (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)>;
G:=Group( (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81), (2,77,38)(3,48,78)(5,80,41)(6,51,81)(8,56,44)(9,54,57)(11,59,47)(12,30,60)(14,62,50)(15,33,63)(17,65,53)(18,36,66)(20,68,29)(21,39,69)(23,71,32)(24,42,72)(26,74,35)(27,45,75)(28,46,37)(31,49,40)(34,52,43)(55,64,73)(58,67,76)(61,70,79), (1,46,58)(2,29,77)(3,39,69)(4,49,61)(5,32,80)(6,42,72)(7,52,64)(8,35,56)(9,45,75)(10,28,67)(11,38,59)(12,48,78)(13,31,70)(14,41,62)(15,51,81)(16,34,73)(17,44,65)(18,54,57)(19,37,76)(20,47,68)(21,30,60)(22,40,79)(23,50,71)(24,33,63)(25,43,55)(26,53,74)(27,36,66), (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) );
G=PermutationGroup([[(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54),(55,58,61,64,67,70,73,76,79),(56,59,62,65,68,71,74,77,80),(57,60,63,66,69,72,75,78,81)], [(2,77,38),(3,48,78),(5,80,41),(6,51,81),(8,56,44),(9,54,57),(11,59,47),(12,30,60),(14,62,50),(15,33,63),(17,65,53),(18,36,66),(20,68,29),(21,39,69),(23,71,32),(24,42,72),(26,74,35),(27,45,75),(28,46,37),(31,49,40),(34,52,43),(55,64,73),(58,67,76),(61,70,79)], [(1,46,58),(2,29,77),(3,39,69),(4,49,61),(5,32,80),(6,42,72),(7,52,64),(8,35,56),(9,45,75),(10,28,67),(11,38,59),(12,48,78),(13,31,70),(14,41,62),(15,51,81),(16,34,73),(17,44,65),(18,54,57),(19,37,76),(20,47,68),(21,30,60),(22,40,79),(23,50,71),(24,33,63),(25,43,55),(26,53,74),(27,36,66)], [(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)]])
C9.6He3 is a maximal subgroup of
He3.2C18 He3.2D9 He3.4D9
51 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 9M | 9N | 27A | ··· | 27R | 27S | ··· | 27AD |
| order | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 27 | ··· | 27 | 27 | ··· | 27 |
| size | 1 | 1 | 1 | 3 | 3 | 9 | 9 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||||
| image | C1 | C3 | C3 | C3 | C9 | C9 | He3 | 3- 1+2 | C9.6He3 |
| kernel | C9.6He3 | C3×C27 | C27⋊C3 | C9○He3 | He3 | 3- 1+2 | C9 | C9 | C1 |
| # reps | 1 | 2 | 4 | 2 | 6 | 12 | 2 | 4 | 18 |
Matrix representation of C9.6He3 ►in GL3(𝔽109) generated by
| 105 | 0 | 0 |
| 0 | 105 | 0 |
| 0 | 0 | 105 |
| 1 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 63 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 23 | 54 | 54 |
| 32 | 54 | 32 |
| 23 | 23 | 32 |
G:=sub<GL(3,GF(109))| [105,0,0,0,105,0,0,0,105],[1,0,0,0,45,0,0,0,63],[0,0,1,1,0,0,0,1,0],[23,32,23,54,54,23,54,32,32] >;
C9.6He3 in GAP, Magma, Sage, TeX
C_9._6{\rm He}_3 % in TeX
G:=Group("C9.6He3"); // GroupNames label
G:=SmallGroup(243,20);
// by ID
G=gap.SmallGroup(243,20);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,1352,457,138]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^3=c^3=1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c^-1,d*c*d^-1=a^3*c>;
// generators/relations
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