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## G = C32.29He3order 243 = 35

### 6th central stem extension by C32 of He3

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C32.29He3, C33.6C32, C32⋊C9.3C3, C3.7(He3.C3), C3.5(He3⋊C3), C3.5(C3.He3), SmallGroup(243,8)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C33 — C32.29He3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.29He3
 Lower central C1 — C32 — C33 — C32.29He3
 Upper central C1 — C32 — C33 — C32.29He3
 Jennings C1 — C32 — C33 — C32.29He3

Generators and relations for C32.29He3
G = < a,b,c,d,e | a3=b3=d3=1, c3=a, e3=b, ab=ba, ac=ca, ad=da, ae=ea, dcd-1=bc=cb, bd=db, be=eb, ece-1=acd-1, ede-1=a-1d >

Smallest permutation representation of C32.29He3
On 81 points
Generators in S81
(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 43 33)(2 44 34)(3 45 35)(4 37 36)(5 38 28)(6 39 29)(7 40 30)(8 41 31)(9 42 32)(10 75 26)(11 76 27)(12 77 19)(13 78 20)(14 79 21)(15 80 22)(16 81 23)(17 73 24)(18 74 25)(46 56 69)(47 57 70)(48 58 71)(49 59 72)(50 60 64)(51 61 65)(52 62 66)(53 63 67)(54 55 68)
(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)
(2 34 44)(3 45 35)(5 28 38)(6 39 29)(8 31 41)(9 42 32)(10 81 20)(11 17 14)(12 25 80)(13 75 23)(15 19 74)(16 78 26)(18 22 77)(21 27 24)(46 49 52)(47 64 63)(48 61 68)(50 67 57)(51 55 71)(53 70 60)(54 58 65)(56 59 62)(66 69 72)(73 79 76)
(1 11 62 43 76 66 33 27 52)(2 25 67 44 18 53 34 74 63)(3 16 55 45 81 68 35 23 54)(4 14 56 37 79 69 36 21 46)(5 19 70 38 12 47 28 77 57)(6 10 58 39 75 71 29 26 48)(7 17 59 40 73 72 30 24 49)(8 22 64 41 15 50 31 80 60)(9 13 61 42 78 65 32 20 51)

G:=sub<Sym(81)| (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), (1,43,33)(2,44,34)(3,45,35)(4,37,36)(5,38,28)(6,39,29)(7,40,30)(8,41,31)(9,42,32)(10,75,26)(11,76,27)(12,77,19)(13,78,20)(14,79,21)(15,80,22)(16,81,23)(17,73,24)(18,74,25)(46,56,69)(47,57,70)(48,58,71)(49,59,72)(50,60,64)(51,61,65)(52,62,66)(53,63,67)(54,55,68), (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), (2,34,44)(3,45,35)(5,28,38)(6,39,29)(8,31,41)(9,42,32)(10,81,20)(11,17,14)(12,25,80)(13,75,23)(15,19,74)(16,78,26)(18,22,77)(21,27,24)(46,49,52)(47,64,63)(48,61,68)(50,67,57)(51,55,71)(53,70,60)(54,58,65)(56,59,62)(66,69,72)(73,79,76), (1,11,62,43,76,66,33,27,52)(2,25,67,44,18,53,34,74,63)(3,16,55,45,81,68,35,23,54)(4,14,56,37,79,69,36,21,46)(5,19,70,38,12,47,28,77,57)(6,10,58,39,75,71,29,26,48)(7,17,59,40,73,72,30,24,49)(8,22,64,41,15,50,31,80,60)(9,13,61,42,78,65,32,20,51)>;

G:=Group( (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), (1,43,33)(2,44,34)(3,45,35)(4,37,36)(5,38,28)(6,39,29)(7,40,30)(8,41,31)(9,42,32)(10,75,26)(11,76,27)(12,77,19)(13,78,20)(14,79,21)(15,80,22)(16,81,23)(17,73,24)(18,74,25)(46,56,69)(47,57,70)(48,58,71)(49,59,72)(50,60,64)(51,61,65)(52,62,66)(53,63,67)(54,55,68), (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), (2,34,44)(3,45,35)(5,28,38)(6,39,29)(8,31,41)(9,42,32)(10,81,20)(11,17,14)(12,25,80)(13,75,23)(15,19,74)(16,78,26)(18,22,77)(21,27,24)(46,49,52)(47,64,63)(48,61,68)(50,67,57)(51,55,71)(53,70,60)(54,58,65)(56,59,62)(66,69,72)(73,79,76), (1,11,62,43,76,66,33,27,52)(2,25,67,44,18,53,34,74,63)(3,16,55,45,81,68,35,23,54)(4,14,56,37,79,69,36,21,46)(5,19,70,38,12,47,28,77,57)(6,10,58,39,75,71,29,26,48)(7,17,59,40,73,72,30,24,49)(8,22,64,41,15,50,31,80,60)(9,13,61,42,78,65,32,20,51) );

G=PermutationGroup([[(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)], [(1,43,33),(2,44,34),(3,45,35),(4,37,36),(5,38,28),(6,39,29),(7,40,30),(8,41,31),(9,42,32),(10,75,26),(11,76,27),(12,77,19),(13,78,20),(14,79,21),(15,80,22),(16,81,23),(17,73,24),(18,74,25),(46,56,69),(47,57,70),(48,58,71),(49,59,72),(50,60,64),(51,61,65),(52,62,66),(53,63,67),(54,55,68)], [(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)], [(2,34,44),(3,45,35),(5,28,38),(6,39,29),(8,31,41),(9,42,32),(10,81,20),(11,17,14),(12,25,80),(13,75,23),(15,19,74),(16,78,26),(18,22,77),(21,27,24),(46,49,52),(47,64,63),(48,61,68),(50,67,57),(51,55,71),(53,70,60),(54,58,65),(56,59,62),(66,69,72),(73,79,76)], [(1,11,62,43,76,66,33,27,52),(2,25,67,44,18,53,34,74,63),(3,16,55,45,81,68,35,23,54),(4,14,56,37,79,69,36,21,46),(5,19,70,38,12,47,28,77,57),(6,10,58,39,75,71,29,26,48),(7,17,59,40,73,72,30,24,49),(8,22,64,41,15,50,31,80,60),(9,13,61,42,78,65,32,20,51)]])

C32.29He3 is a maximal subgroup of   C32⋊C9.C6  C33.(C3×S3)  C3.(He3⋊S3)

35 conjugacy classes

 class 1 3A ··· 3H 3I 3J 9A ··· 9X order 1 3 ··· 3 3 3 9 ··· 9 size 1 1 ··· 1 9 9 9 ··· 9

35 irreducible representations

 dim 1 1 3 3 3 3 type + image C1 C3 He3 He3.C3 He3⋊C3 C3.He3 kernel C32.29He3 C32⋊C9 C32 C3 C3 C3 # reps 1 8 2 6 6 12

Matrix representation of C32.29He3 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 5 0 0 0 0 0 0 16 6 0 0 0 0 0 3 1 0 0 0 11 10 0
,
 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 16 7 0 0 0 0 15 0 11
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 14 0 9

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,0,0,0,0,0,0,16,0,11,0,0,0,6,3,10,0,0,0,0,1,0],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,16,15,0,0,0,0,7,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,14,0,0,0,0,6,0,0,0,0,0,0,9] >;

C32.29He3 in GAP, Magma, Sage, TeX

C_3^2._{29}{\rm He}_3
% in TeX

G:=Group("C3^2.29He3");
// GroupNames label

G:=SmallGroup(243,8);
// by ID

G=gap.SmallGroup(243,8);
# by ID

G:=PCGroup([5,-3,3,-3,-3,3,405,121,96,542,457]);
// Polycyclic

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

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