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## G = C3×C32⋊C9order 243 = 35

### Direct product of C3 and C32⋊C9

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

Aliases: C3×C32⋊C9, C332C9, C34.2C3, C32.23He3, C33.23C32, C32.15C33, C32.103- 1+2, (C3×C9)⋊5C32, C322(C3×C9), (C32×C9)⋊2C3, C3.1(C3×He3), C3.1(C32×C9), C3.1(C3×3- 1+2), SmallGroup(243,32)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3 — C3×C32⋊C9
 Chief series C1 — C3 — C32 — C33 — C34 — C3×C32⋊C9
 Lower central C1 — C3 — C3×C32⋊C9
 Upper central C1 — C33 — C3×C32⋊C9
 Jennings C1 — C3 — C32 — C3×C32⋊C9

Generators and relations for C3×C32⋊C9
G = < a,b,c,d | a3=b3=c3=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 288 in 144 conjugacy classes, 72 normal (7 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, C33, C33, C33, C32⋊C9, C32×C9, C34, C3×C32⋊C9
Quotients: C1, C3, C9, C32, C3×C9, He3, 3- 1+2, C33, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C3×C32⋊C9

Smallest permutation representation of C3×C32⋊C9
On 81 points
Generators in S81
(1 56 65)(2 57 66)(3 58 67)(4 59 68)(5 60 69)(6 61 70)(7 62 71)(8 63 72)(9 55 64)(10 38 30)(11 39 31)(12 40 32)(13 41 33)(14 42 34)(15 43 35)(16 44 36)(17 45 28)(18 37 29)(19 81 52)(20 73 53)(21 74 54)(22 75 46)(23 76 47)(24 77 48)(25 78 49)(26 79 50)(27 80 51)
(1 51 28)(2 63 69)(3 41 26)(4 54 31)(5 57 72)(6 44 20)(7 48 34)(8 60 66)(9 38 23)(10 47 64)(11 59 21)(12 37 35)(13 50 67)(14 62 24)(15 40 29)(16 53 70)(17 56 27)(18 43 32)(19 78 46)(22 81 49)(25 75 52)(30 76 55)(33 79 58)(36 73 61)(39 68 74)(42 71 77)(45 65 80)
(1 74 14)(2 75 15)(3 76 16)(4 77 17)(5 78 18)(6 79 10)(7 80 11)(8 81 12)(9 73 13)(19 32 72)(20 33 64)(21 34 65)(22 35 66)(23 36 67)(24 28 68)(25 29 69)(26 30 70)(27 31 71)(37 60 49)(38 61 50)(39 62 51)(40 63 52)(41 55 53)(42 56 54)(43 57 46)(44 58 47)(45 59 48)
(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)| (1,56,65)(2,57,66)(3,58,67)(4,59,68)(5,60,69)(6,61,70)(7,62,71)(8,63,72)(9,55,64)(10,38,30)(11,39,31)(12,40,32)(13,41,33)(14,42,34)(15,43,35)(16,44,36)(17,45,28)(18,37,29)(19,81,52)(20,73,53)(21,74,54)(22,75,46)(23,76,47)(24,77,48)(25,78,49)(26,79,50)(27,80,51), (1,51,28)(2,63,69)(3,41,26)(4,54,31)(5,57,72)(6,44,20)(7,48,34)(8,60,66)(9,38,23)(10,47,64)(11,59,21)(12,37,35)(13,50,67)(14,62,24)(15,40,29)(16,53,70)(17,56,27)(18,43,32)(19,78,46)(22,81,49)(25,75,52)(30,76,55)(33,79,58)(36,73,61)(39,68,74)(42,71,77)(45,65,80), (1,74,14)(2,75,15)(3,76,16)(4,77,17)(5,78,18)(6,79,10)(7,80,11)(8,81,12)(9,73,13)(19,32,72)(20,33,64)(21,34,65)(22,35,66)(23,36,67)(24,28,68)(25,29,69)(26,30,70)(27,31,71)(37,60,49)(38,61,50)(39,62,51)(40,63,52)(41,55,53)(42,56,54)(43,57,46)(44,58,47)(45,59,48), (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( (1,56,65)(2,57,66)(3,58,67)(4,59,68)(5,60,69)(6,61,70)(7,62,71)(8,63,72)(9,55,64)(10,38,30)(11,39,31)(12,40,32)(13,41,33)(14,42,34)(15,43,35)(16,44,36)(17,45,28)(18,37,29)(19,81,52)(20,73,53)(21,74,54)(22,75,46)(23,76,47)(24,77,48)(25,78,49)(26,79,50)(27,80,51), (1,51,28)(2,63,69)(3,41,26)(4,54,31)(5,57,72)(6,44,20)(7,48,34)(8,60,66)(9,38,23)(10,47,64)(11,59,21)(12,37,35)(13,50,67)(14,62,24)(15,40,29)(16,53,70)(17,56,27)(18,43,32)(19,78,46)(22,81,49)(25,75,52)(30,76,55)(33,79,58)(36,73,61)(39,68,74)(42,71,77)(45,65,80), (1,74,14)(2,75,15)(3,76,16)(4,77,17)(5,78,18)(6,79,10)(7,80,11)(8,81,12)(9,73,13)(19,32,72)(20,33,64)(21,34,65)(22,35,66)(23,36,67)(24,28,68)(25,29,69)(26,30,70)(27,31,71)(37,60,49)(38,61,50)(39,62,51)(40,63,52)(41,55,53)(42,56,54)(43,57,46)(44,58,47)(45,59,48), (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([[(1,56,65),(2,57,66),(3,58,67),(4,59,68),(5,60,69),(6,61,70),(7,62,71),(8,63,72),(9,55,64),(10,38,30),(11,39,31),(12,40,32),(13,41,33),(14,42,34),(15,43,35),(16,44,36),(17,45,28),(18,37,29),(19,81,52),(20,73,53),(21,74,54),(22,75,46),(23,76,47),(24,77,48),(25,78,49),(26,79,50),(27,80,51)], [(1,51,28),(2,63,69),(3,41,26),(4,54,31),(5,57,72),(6,44,20),(7,48,34),(8,60,66),(9,38,23),(10,47,64),(11,59,21),(12,37,35),(13,50,67),(14,62,24),(15,40,29),(16,53,70),(17,56,27),(18,43,32),(19,78,46),(22,81,49),(25,75,52),(30,76,55),(33,79,58),(36,73,61),(39,68,74),(42,71,77),(45,65,80)], [(1,74,14),(2,75,15),(3,76,16),(4,77,17),(5,78,18),(6,79,10),(7,80,11),(8,81,12),(9,73,13),(19,32,72),(20,33,64),(21,34,65),(22,35,66),(23,36,67),(24,28,68),(25,29,69),(26,30,70),(27,31,71),(37,60,49),(38,61,50),(39,62,51),(40,63,52),(41,55,53),(42,56,54),(43,57,46),(44,58,47),(45,59,48)], [(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)]])

C3×C32⋊C9 is a maximal subgroup of   C33⋊C18  C33⋊D9  C336D9

99 conjugacy classes

 class 1 3A ··· 3Z 3AA ··· 3AR 9A ··· 9BB order 1 3 ··· 3 3 ··· 3 9 ··· 9 size 1 1 ··· 1 3 ··· 3 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 1 3 3 type + image C1 C3 C3 C3 C9 He3 3- 1+2 kernel C3×C32⋊C9 C32⋊C9 C32×C9 C34 C33 C32 C32 # reps 1 18 6 2 54 6 12

Matrix representation of C3×C32⋊C9 in GL5(𝔽19)

 11 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 1 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 11
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 9 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(19))| [11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[9,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C3×C32⋊C9 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes C_9
% in TeX

G:=Group("C3xC3^2:C9");
// GroupNames label

G:=SmallGroup(243,32);
// by ID

G=gap.SmallGroup(243,32);
# by ID

G:=PCGroup([5,-3,3,3,-3,3,405,301]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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