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## G = C32×C7⋊C3order 189 = 33·7

### Direct product of C32 and C7⋊C3

Aliases: C32×C7⋊C3, C7⋊C33, C21⋊C32, (C3×C21)⋊3C3, SmallGroup(189,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C32×C7⋊C3
 Chief series C1 — C7 — C7⋊C3 — C3×C7⋊C3 — C32×C7⋊C3
 Lower central C7 — C32×C7⋊C3
 Upper central C1 — C32

Generators and relations for C32×C7⋊C3
G = < a,b,c,d | a3=b3=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 188 in 56 conjugacy classes, 34 normal (5 characteristic)
C1, C3, C3, C7, C32, C32, C7⋊C3, C21, C33, C3×C7⋊C3, C3×C21, C32×C7⋊C3
Quotients: C1, C3, C32, C7⋊C3, C33, C3×C7⋊C3, C32×C7⋊C3

Smallest permutation representation of C32×C7⋊C3
On 63 points
Generators in S63
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 43 36)(9 44 37)(10 45 38)(11 46 39)(12 47 40)(13 48 41)(14 49 42)(15 50 22)(16 51 23)(17 52 24)(18 53 25)(19 54 26)(20 55 27)(21 56 28)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)
(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)
(1 8 15)(2 10 19)(3 12 16)(4 14 20)(5 9 17)(6 11 21)(7 13 18)(22 29 36)(23 31 40)(24 33 37)(25 35 41)(26 30 38)(27 32 42)(28 34 39)(43 50 57)(44 52 61)(45 54 58)(46 56 62)(47 51 59)(48 53 63)(49 55 60)

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

G:=Group( (1,57,29)(2,58,30)(3,59,31)(4,60,32)(5,61,33)(6,62,34)(7,63,35)(8,43,36)(9,44,37)(10,45,38)(11,46,39)(12,47,40)(13,48,41)(14,49,42)(15,50,22)(16,51,23)(17,52,24)(18,53,25)(19,54,26)(20,55,27)(21,56,28), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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), (1,8,15)(2,10,19)(3,12,16)(4,14,20)(5,9,17)(6,11,21)(7,13,18)(22,29,36)(23,31,40)(24,33,37)(25,35,41)(26,30,38)(27,32,42)(28,34,39)(43,50,57)(44,52,61)(45,54,58)(46,56,62)(47,51,59)(48,53,63)(49,55,60) );

G=PermutationGroup([[(1,57,29),(2,58,30),(3,59,31),(4,60,32),(5,61,33),(6,62,34),(7,63,35),(8,43,36),(9,44,37),(10,45,38),(11,46,39),(12,47,40),(13,48,41),(14,49,42),(15,50,22),(16,51,23),(17,52,24),(18,53,25),(19,54,26),(20,55,27),(21,56,28)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56)], [(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)], [(1,8,15),(2,10,19),(3,12,16),(4,14,20),(5,9,17),(6,11,21),(7,13,18),(22,29,36),(23,31,40),(24,33,37),(25,35,41),(26,30,38),(27,32,42),(28,34,39),(43,50,57),(44,52,61),(45,54,58),(46,56,62),(47,51,59),(48,53,63),(49,55,60)]])

C32×C7⋊C3 is a maximal subgroup of   C324F7

45 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3Z 7A 7B 21A ··· 21P order 1 3 ··· 3 3 ··· 3 7 7 21 ··· 21 size 1 1 ··· 1 7 ··· 7 3 3 3 ··· 3

45 irreducible representations

 dim 1 1 1 3 3 type + image C1 C3 C3 C7⋊C3 C3×C7⋊C3 kernel C32×C7⋊C3 C3×C7⋊C3 C3×C21 C32 C3 # reps 1 24 2 2 16

Matrix representation of C32×C7⋊C3 in GL4(𝔽43) generated by

 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 42 18 1 0 0 18 1 0 42 19 1
,
 36 0 0 0 0 28 6 21 0 6 0 0 0 6 6 15
G:=sub<GL(4,GF(43))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,42,0,42,0,18,18,19,0,1,1,1],[36,0,0,0,0,28,6,6,0,6,0,6,0,21,0,15] >;

C32×C7⋊C3 in GAP, Magma, Sage, TeX

C_3^2\times C_7\rtimes C_3
% in TeX

G:=Group("C3^2xC7:C3");
// GroupNames label

G:=SmallGroup(189,12);
// by ID

G=gap.SmallGroup(189,12);
# by ID

G:=PCGroup([4,-3,-3,-3,-7,867]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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