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G = C32.24He3order 243 = 35

1st central stem extension by C32 of He3

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

Aliases: C32.24He3, C33.1C32, C3.4C3wrC3, C32:C9:1C3, (C3xHe3):1C3, C3.3(He3:C3), SmallGroup(243,3)

Series: Derived Chief Lower central Upper central Jennings

C1C33 — C32.24He3
C1C3C32C33C3xHe3 — C32.24He3
C1C32C33 — C32.24He3
C1C32C33 — C32.24He3
C1C32C33 — C32.24He3

Generators and relations for C32.24He3
 G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, dcd-1=bc=cb, bd=db, be=eb, ece-1=acd-1, ede-1=a-1d >

Subgroups: 207 in 47 conjugacy classes, 12 normal (4 characteristic)
Quotients: C1, C3, C32, He3, C3wrC3, He3:C3, C32.24He3
9C3
9C3
9C3
9C3
9C3
9C3
9C3
3C32
3C32
3C32
3C32
3C32
3C32
3C32
3C32
3C32
3C32
9C32
9C9
9C32
9C32
9C9
9C32
9C32
9C32
3He3
3C33
3C33
3He3
3C3xC9
3C3xC9
3He3
3He3
3He3
3He3

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

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

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

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

C32.24He3 is a maximal subgroup of   C3.C3wrS3  C32:C9.S3  (C3xHe3):S3

35 conjugacy classes

class 1 3A···3H3I···3V9A···9L
order13···33···39···9
size11···19···99···9

35 irreducible representations

dim111333
type+
imageC1C3C3He3C3wrC3He3:C3
kernelC32.24He3C32:C9C3xHe3C32C3C3
# reps14421212

Matrix representation of C32.24He3 in GL6(F19)

1100000
0110000
0011000
0001100
0000110
0000011
,
700000
070000
007000
000100
000010
000001
,
700000
1810000
7011000
000700
000070
00011161
,
160000
0181000
0180000
0001110
0000110
0008157
,
946000
9010000
15410000
0007010
00015816
00013114

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[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],[7,18,7,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,11,0,0,0,0,7,16,0,0,0,0,0,1],[1,0,0,0,0,0,6,18,18,0,0,0,0,1,0,0,0,0,0,0,0,1,0,8,0,0,0,11,11,15,0,0,0,0,0,7],[9,9,15,0,0,0,4,0,4,0,0,0,6,10,10,0,0,0,0,0,0,7,15,13,0,0,0,0,8,11,0,0,0,10,16,4] >;

C32.24He3 in GAP, Magma, Sage, TeX

C_3^2._{24}{\rm He}_3
% in TeX

G:=Group("C3^2.24He3");
// GroupNames label

G:=SmallGroup(243,3);
// by ID

G=gap.SmallGroup(243,3);
# by ID

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

G:=Group<a,b,c,d,e|a^3=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,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

Export

Subgroup lattice of C32.24He3 in TeX

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