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## G = C32×F7order 378 = 2·33·7

### Direct product of C32 and F7

Aliases: C32×F7, D7⋊C33, C7⋊(C32×C6), (C3×C21)⋊7C6, C212(C3×C6), (C3×D7)⋊C32, (C32×D7)⋊3C3, C7⋊C3⋊(C3×C6), (C3×C7⋊C3)⋊4C6, (C32×C7⋊C3)⋊3C2, SmallGroup(378,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C32×F7
 Chief series C1 — C7 — C21 — C3×C21 — C32×C7⋊C3 — C32×F7
 Lower central C7 — C32×F7
 Upper central C1 — C32

Generators and relations for C32×F7
G = < a,b,c,d | a3=b3=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 412 in 112 conjugacy classes, 62 normal (8 characteristic)
C1, C2, C3, C3, C6, C7, C32, C32, D7, C3×C6, C7⋊C3, C21, C33, F7, C3×D7, C32×C6, C3×C7⋊C3, C3×C21, C3×F7, C32×D7, C32×C7⋊C3, C32×F7
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, F7, C32×C6, C3×F7, C32×F7

Smallest permutation representation of C32×F7
On 63 points
Generators in S63
(1 43 22)(2 44 23)(3 45 24)(4 46 25)(5 47 26)(6 48 27)(7 49 28)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)
(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 15 8)(2 18 10 7 19 13)(3 21 12 6 16 11)(4 17 14 5 20 9)(22 36 29)(23 39 31 28 40 34)(24 42 33 27 37 32)(25 38 35 26 41 30)(43 57 50)(44 60 52 49 61 55)(45 63 54 48 58 53)(46 59 56 47 62 51)

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

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

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

63 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Z 6A ··· 6Z 7 21A ··· 21H order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 7 21 ··· 21 size 1 7 1 ··· 1 7 ··· 7 7 ··· 7 6 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 6 6 type + + + image C1 C2 C3 C3 C6 C6 F7 C3×F7 kernel C32×F7 C32×C7⋊C3 C3×F7 C32×D7 C3×C7⋊C3 C3×C21 C32 C3 # reps 1 1 24 2 24 2 1 8

Matrix representation of C32×F7 in GL7(𝔽43)

 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 42 0 1 0 0 0 0 42 0 0 1 0 0 0 42 0 0 0 1 0 0 42 0 0 0 0 1 0 42 0 0 0 0 0 1 42
,
 42 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0

G:=sub<GL(7,GF(43))| [6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,42,42,42,42,42,42],[42,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,0,0,36,0,0,0,0,0,0,0,0,0,36,0,0,36,0,0,0,0,0,0,0,0,0,36,0,0] >;

C32×F7 in GAP, Magma, Sage, TeX

C_3^2\times F_7
% in TeX

G:=Group("C3^2xF7");
// GroupNames label

G:=SmallGroup(378,47);
// by ID

G=gap.SmallGroup(378,47);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,8104,2709]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^7=d^6=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^5>;
// generators/relations

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