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## G = C32.27He3order 243 = 35

### 4th central stem extension by C32 of He3

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

Aliases: C32.27He3, C33.4C32, C3.7C3≀C3, C32⋊C93C3, (C3×He3).2C3, C3.6(He3.C3), C3.4(He3⋊C3), SmallGroup(243,6)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C33 — C32.27He3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.27He3
 Lower central C1 — C32 — C33 — C32.27He3
 Upper central C1 — C32 — C33 — C32.27He3
 Jennings C1 — C32 — C33 — C32.27He3

Generators and relations for C32.27He3
G = < a,b,c,d,e | a3=b3=c3=d3=1, e3=b-1, ab=ba, ac=ca, dad-1=ab-1, eae-1=ac-1, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=ab-1c-1d >

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

C32.27He3 is a maximal subgroup of   C3.3C3≀S3  (C3×He3).C6  C32⋊C96S3

35 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3P 9A ··· 9R order 1 3 ··· 3 3 ··· 3 9 ··· 9 size 1 1 ··· 1 9 ··· 9 9 ··· 9

35 irreducible representations

 dim 1 1 1 3 3 3 3 type + image C1 C3 C3 He3 C3≀C3 He3.C3 He3⋊C3 kernel C32.27He3 C32⋊C9 C3×He3 C32 C3 C3 C3 # reps 1 6 2 2 6 12 6

Matrix representation of C32.27He3 in GL6(𝔽19)

 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 7 0 0 0 0 0 9 1 0 0 0 0 14 0 11
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 7 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 9 1 0 0 0 0 17 0 1
,
 12 12 18 0 0 0 12 8 8 0 0 0 18 8 18 0 0 0 0 0 0 9 13 0 0 0 0 17 10 1 0 0 0 0 2 0

G:=sub<GL(6,GF(19))| [0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,9,14,0,0,0,0,1,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,9,17,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,18,0,0,0,12,8,8,0,0,0,18,8,18,0,0,0,0,0,0,9,17,0,0,0,0,13,10,2,0,0,0,0,1,0] >;

C32.27He3 in GAP, Magma, Sage, TeX

C_3^2._{27}{\rm He}_3
% in TeX

G:=Group("C3^2.27He3");
// GroupNames label

G:=SmallGroup(243,6);
// by ID

G=gap.SmallGroup(243,6);
# by ID

G:=PCGroup([5,-3,3,-3,-3,3,121,186,542,457]);
// Polycyclic

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

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