C3^2.23C3^3 - GroupNames

G = C₃².₂₃C₃³ order 243 = 3⁵

4^th central stem extension by C₃² of C₃³

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₃².₈He₃, C₃³.₂₉C₃², C₃².₂₃C₃³, C₃²⋊C₉⋊₈C₃, (C₃²×C₉)⋊₄C₃, C₃.₆(C₃×He₃), (C₃×He₃).₅C₃, (C₃×C₉).₆C₃², C₃.₅(C₉○He₃), (C₃×3_-¹⁺²)⋊₄C₃, SmallGroup(243,40)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₃² — C₃².₂₃C₃³

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃²×C₉ — C₃².₂₃C₃³

Lower central

C₁ — C₃² — C₃².₂₃C₃³

Upper central

C₁ — C₃² — C₃².₂₃C₃³

Jennings

C₁ — C₃ — C₃² — C₃².₂₃C₃³

Generators and relations for C₃².₂₃C₃³
G = < a,b,c,d,e | a³=b³=c³=d³=1, e³=a, ab=ba, dcd^-1=ac=ca, ad=da, ae=ea, ece^-1=bc=cb, bd=db, be=eb, de=ed >

Subgroups: 180 in 78 conjugacy classes, 36 normal (7 characteristic)
C₁, C₃, C₃, C₃, C₉, C₃², C₃², C₃², C₃×C₉, C₃×C₉, He₃, 3_-¹⁺², C₃³, C₃³, C₃²⋊C₉, C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², C₃².₂₃C₃³
Quotients: C₁, C₃, C₃², He₃, C₃³, C₃×He₃, C₉○He₃, C₃².₂₃C₃³

Smallest permutation representation of C₃².₂₃C₃³
►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 44 34)(2 45 35)(3 37 36)(4 38 28)(5 39 29)(6 40 30)(7 41 31)(8 42 32)(9 43 33)(10 74 20)(11 75 21)(12 76 22)(13 77 23)(14 78 24)(15 79 25)(16 80 26)(17 81 27)(18 73 19)(46 66 56)(47 67 57)(48 68 58)(49 69 59)(50 70 60)(51 71 61)(52 72 62)(53 64 63)(54 65 55)

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

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

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

C₃².₂₃C₃³ is a maximal subgroup of (C₃²×C₉)⋊S₃ (C₃²×C₉)⋊₈S₃ (C₃²×C₉)⋊C₆

51 conjugacy classes

class	1	3A	···	3H	3I	···	3N	3O	···	3T	9A	···	9R	9S	···	9AD
order	1	3	···	3	3	···	3	3	···	3	9	···	9	9	···	9
size	1	1	···	1	3	···	3	9	···	9	3	···	3	9	···	9

51 irreducible representations

dim	1	1	1	1	1	3	3
type	+
image	C₁	C₃	C₃	C₃	C₃	He₃	C₉○He₃
kernel	C₃².₂₃C₃³	C₃²⋊C₉	C₃²×C₉	C₃×He₃	C₃×3_-¹⁺²	C₃²	C₃
# reps	1	18	2	2	4	6	18

Matrix representation of C₃².₂₃C₃³ ►in GL₆(𝔽₁₉)

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	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

0	0	4	0	0	0
6	0	0	0	0	0
0	4	0	0	0	0
0	0	0	1	0	0
0	0	0	0	11	0
0	0	0	0	0	7

0	0	9	0	0	0
6	0	0	0	0	0
0	6	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
0	0	1	0	0	0
7	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(19))| [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],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,6,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[0,6,0,0,0,0,0,0,6,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C₃².₂₃C₃³ in GAP, Magma, Sage, TeX

C_3^2._{23}C_3^3

% in TeX

G:=Group("C3^2.23C3^3");

// GroupNames label

G:=SmallGroup(243,40);

// by ID

G=gap.SmallGroup(243,40);

# by ID

G:=PCGroup([5,-3,3,3,-3,3,301,1352,57]);

// Polycyclic

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

// generators/relations

G = C32.23C33 order 243 = 35

4th central stem extension by C32 of C33

G = C₃².₂₃C₃³ order 243 = 3⁵

4^th central stem extension by C₃² of C₃³