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## G = C3×S3×C7⋊C3order 378 = 2·33·7

### Direct product of C3, S3 and C7⋊C3

Aliases: C3×S3×C7⋊C3, (S3×C21)⋊C3, C214(C3×S3), C213(C3×C6), (S3×C7)⋊C32, C72(S3×C32), (C3×C21)⋊10C6, C3⋊(C6×C7⋊C3), (C3×C7⋊C3)⋊5C6, (C32×C7⋊C3)⋊4C2, C323(C2×C7⋊C3), SmallGroup(378,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C3×S3×C7⋊C3
 Chief series C1 — C7 — C21 — C3×C21 — C32×C7⋊C3 — C3×S3×C7⋊C3
 Lower central C21 — C3×S3×C7⋊C3
 Upper central C1 — C3

Generators and relations for C3×S3×C7⋊C3
G = < a,b,c,d,e | a3=b3=c2=d7=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 332 in 64 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3, C3, S3, C6, C7, C32, C32, C14, C3×S3, C3×S3, C3×C6, C7⋊C3, C7⋊C3, C21, C21, C33, C2×C7⋊C3, S3×C7, C42, S3×C32, C3×C7⋊C3, C3×C7⋊C3, C3×C7⋊C3, C3×C21, S3×C7⋊C3, C6×C7⋊C3, S3×C21, C32×C7⋊C3, C3×S3×C7⋊C3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C7⋊C3, C2×C7⋊C3, S3×C32, C3×C7⋊C3, S3×C7⋊C3, C6×C7⋊C3, C3×S3×C7⋊C3

Smallest permutation representation of C3×S3×C7⋊C3
On 42 points
Generators in S42
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)

G:=sub<Sym(42)| (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)>;

G:=Group( (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41) );

G=PermutationGroup([[(1,8,15),(2,9,16),(3,10,17),(4,11,18),(5,12,19),(6,13,20),(7,14,21),(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42)], [(1,8,15),(2,9,16),(3,10,17),(4,11,18),(5,12,19),(6,13,20),(7,14,21),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34),(37,38,40),(39,42,41)]])

45 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3Q 6A 6B 6C ··· 6H 7A 7B 14A 14B 21A 21B 21C 21D 21E ··· 21J 42A 42B 42C 42D order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 7 7 14 14 21 21 21 21 21 ··· 21 42 42 42 42 size 1 3 1 1 2 2 2 7 ··· 7 14 ··· 14 3 3 21 ··· 21 3 3 9 9 3 3 3 3 6 ··· 6 9 9 9 9

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 6 6 type + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 C7⋊C3 C2×C7⋊C3 C3×C7⋊C3 C6×C7⋊C3 S3×C7⋊C3 C3×S3×C7⋊C3 kernel C3×S3×C7⋊C3 C32×C7⋊C3 S3×C7⋊C3 S3×C21 C3×C7⋊C3 C3×C21 C3×C7⋊C3 C7⋊C3 C21 C3×S3 C32 S3 C3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 2 2 4 4 2 4

Matrix representation of C3×S3×C7⋊C3 in GL5(𝔽43)

 6 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 6 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 18 19 1 0 0 1 0 0 0 0 0 1 0
,
 36 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 24 42 42 0 0 0 1 0

G:=sub<GL(5,GF(43))| [6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,18,1,0,0,0,19,0,1,0,0,1,0,0],[36,0,0,0,0,0,36,0,0,0,0,0,1,24,0,0,0,0,42,1,0,0,0,42,0] >;

C3×S3×C7⋊C3 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_7\rtimes C_3
% in TeX

G:=Group("C3xS3xC7:C3");
// GroupNames label

G:=SmallGroup(378,48);
// by ID

G=gap.SmallGroup(378,48);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,723,1359]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^7=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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