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## G = C3×C3⋊F7order 378 = 2·33·7

### Direct product of C3 and C3⋊F7

Aliases: C3×C3⋊F7, D21⋊C32, C323F7, C3⋊(C3×F7), C7⋊(S3×C32), (C3×D21)⋊C3, C212(C3×S3), (C3×C21)⋊2C6, C211(C3×C6), C7⋊C3⋊(C3×S3), (C3×C7⋊C3)⋊3C6, (C3×C7⋊C3)⋊4S3, (C32×C7⋊C3)⋊1C2, SmallGroup(378,49)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C3⋊F7
G = < a,b,c,d | a3=b3=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 380 in 64 conjugacy classes, 22 normal (13 characteristic)
C1, C2, C3, C3, S3, C6, C7, C32, C32, D7, C3×S3, C3×C6, C7⋊C3, C7⋊C3, C21, C21, C33, F7, C3×D7, D21, S3×C32, C3×C7⋊C3, C3×C7⋊C3, C3×C7⋊C3, C3×C21, C3×F7, C3⋊F7, C3×D21, C32×C7⋊C3, C3×C3⋊F7
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, F7, S3×C32, C3×F7, C3⋊F7, C3×C3⋊F7

Smallest permutation representation of C3×C3⋊F7
On 42 points
Generators in S42
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)
(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 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)
(1 29 20 22 13 36)(2 32 15 28 10 41)(3 35 17 27 14 39)(4 31 19 26 11 37)(5 34 21 25 8 42)(6 30 16 24 12 40)(7 33 18 23 9 38)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 6 6 6 6 type + + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 F7 C3×F7 C3⋊F7 C3×C3⋊F7 kernel C3×C3⋊F7 C32×C7⋊C3 C3⋊F7 C3×D21 C3×C7⋊C3 C3×C21 C3×C7⋊C3 C7⋊C3 C21 C32 C3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 1 2 2 4

Matrix representation of C3×C3⋊F7 in GL8(𝔽43)

 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 42 1 0 0 0 0 0 0 42 0 1 0 0 0 0 0 42 0 0 1 0 0 0 0 42 0 0 0 1 0 0 0 42 0 0 0 0 1 0 0 42 0 0 0 0 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 36 42 0 0 1 1 36 0 42 0 0 0 37 0 42 1 42 0 0 0 0 1 42 1 0 36 0 0 0 1 0 37 42 42 0 0 1 37 42 0 0 42

G:=sub<GL(8,GF(43))| [36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,42,42,42,42,42,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,37,0,0,1,0,0,0,1,0,1,1,37,0,0,0,36,42,42,0,42,0,0,1,0,1,1,37,0,0,0,36,42,42,0,42,0,0,0,42,0,0,36,42,42] >;

C3×C3⋊F7 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes F_7
% in TeX

G:=Group("C3xC3:F7");
// GroupNames label

G:=SmallGroup(378,49);
// by ID

G=gap.SmallGroup(378,49);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,723,8104,1359]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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