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## G = C32×D7order 126 = 2·32·7

### Direct product of C32 and D7

Aliases: C32×D7, C216C6, C73(C3×C6), (C3×C21)⋊3C2, SmallGroup(126,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C32×D7
 Chief series C1 — C7 — C21 — C3×C21 — C32×D7
 Lower central C7 — C32×D7
 Upper central C1 — C32

Generators and relations for C32×D7
G = < a,b,c,d | a3=b3=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C32×D7
On 63 points
Generators in S63
(1 62 34)(2 63 35)(3 57 29)(4 58 30)(5 59 31)(6 60 32)(7 61 33)(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 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)

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

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

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

C32×D7 is a maximal subgroup of   C32.F7  D7⋊He3

45 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 7A 7B 7C 21A ··· 21X order 1 2 3 ··· 3 6 ··· 6 7 7 7 21 ··· 21 size 1 7 1 ··· 1 7 ··· 7 2 2 2 2 ··· 2

45 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 D7 C3×D7 kernel C32×D7 C3×C21 C3×D7 C21 C32 C3 # reps 1 1 8 8 3 24

Matrix representation of C32×D7 in GL3(𝔽43) generated by

 36 0 0 0 6 0 0 0 6
,
 6 0 0 0 1 0 0 0 1
,
 1 0 0 0 42 1 0 33 9
,
 42 0 0 0 42 0 0 33 1
G:=sub<GL(3,GF(43))| [36,0,0,0,6,0,0,0,6],[6,0,0,0,1,0,0,0,1],[1,0,0,0,42,33,0,1,9],[42,0,0,0,42,33,0,0,1] >;

C32×D7 in GAP, Magma, Sage, TeX

C_3^2\times D_7
% in TeX

G:=Group("C3^2xD7");
// GroupNames label

G:=SmallGroup(126,11);
// by ID

G=gap.SmallGroup(126,11);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,1731]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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