C3xS3xC7:C3 - GroupNames

G = C₃×S₃×C₇⋊C₃ order 378 = 2·3³·7

Direct product of C₃, S₃ and C₇⋊C₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃×S₃×C₇⋊C₃, (S₃×C₂₁)⋊C₃, C₂₁⋊₄(C₃×S₃), C₂₁⋊₃(C₃×C₆), (S₃×C₇)⋊C₃², C₇⋊₂(S₃×C₃²), (C₃×C₂₁)⋊₁₀C₆, C₃⋊(C₆×C₇⋊C₃), (C₃×C₇⋊C₃)⋊₅C₆, (C₃²×C₇⋊C₃)⋊₄C₂, C₃²⋊₃(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

G = C3×S3×C7⋊C3 order 378 = 2·33·7

Direct product of C3, S3 and C7⋊C3

G = C₃×S₃×C₇⋊C₃ order 378 = 2·3³·7

Direct product of C₃, S₃ and C₇⋊C₃