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G = C2×S3×C7⋊C3  order 252 = 22·32·7

Direct product of C2, S3 and C7⋊C3

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

Aliases: C2×S3×C7⋊C3, C42⋊3C6, C7⋊3(S3×C6), (S3×C14)⋊C3, (S3×C7)⋊2C6, C14⋊2(C3×S3), C21⋊4(C2×C6), C6⋊(C2×C7⋊C3), C3⋊(C22×C7⋊C3), (C6×C7⋊C3)⋊3C2, (C3×C7⋊C3)⋊4C22, SmallGroup(252,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C21 — C2×S3×C7⋊C3
Chief series
C1 — C7 — C21 — C3×C7⋊C3 — S3×C7⋊C3 — C2×S3×C7⋊C3
Lower central
C21 — C2×S3×C7⋊C3
Upper central
C1 — C2

Generators and relations for C2×S3×C7⋊C3
 G = < a,b,c,d,e | a2=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 >

C1
C2
3C2
3C2
C3
7C3
14C3
C7
3C22
S3
S3
C6
7C6
14C6
21C6
21C6
7C32
C14
3C14
3C14
C21
C7⋊C3
2C7⋊C3
D6
21C2×C6
7C3×S3
7C3×S3
7C3×C6
3C2×C14
C2×C7⋊C3
C42
S3×C7
S3×C7
2C2×C7⋊C3
3C2×C7⋊C3
3C2×C7⋊C3
C3×C7⋊C3
7S3×C6
S3×C14
3C22×C7⋊C3
S3×C7⋊C3
S3×C7⋊C3
C6×C7⋊C3
C2×S3×C7⋊C3

Character table of C2×S3×C7⋊C3

 class 12A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I7A7B14A14B14C14D14E14F21A21B42A42B
 size 11332771414277141421212121333399996666
ρ1111111111111111111111111111111    trivial
ρ21-11-111111-1-1-1-1-11-11-111-1-11-11-111-1-1    linear of order 2
ρ311-1-11111111111-1-1-1-11111-1-1-1-11111    linear of order 2
ρ41-1-1111111-1-1-1-1-1-11-1111-1-1-11-1111-1-1    linear of order 2
ρ51-1-111ζ3ζ32ζ32ζ3-1ζ65ζ6ζ6ζ65ζ65ζ32ζ6ζ311-1-1-11-1111-1-1    linear of order 6
ρ61-11-11ζ3ζ32ζ32ζ3-1ζ65ζ6ζ6ζ65ζ3ζ6ζ32ζ6511-1-11-11-111-1-1    linear of order 6
ρ711111ζ32ζ3ζ3ζ321ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32111111111111    linear of order 3
ρ811-1-11ζ3ζ32ζ32ζ31ζ3ζ32ζ32ζ3ζ65ζ6ζ6ζ651111-1-1-1-11111    linear of order 6
ρ911111ζ3ζ32ζ32ζ31ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3111111111111    linear of order 3
ρ101-11-11ζ32ζ3ζ3ζ32-1ζ6ζ65ζ65ζ6ζ32ζ65ζ3ζ611-1-11-11-111-1-1    linear of order 6
ρ111-1-111ζ32ζ3ζ3ζ32-1ζ6ζ65ζ65ζ6ζ6ζ3ζ65ζ3211-1-1-11-1111-1-1    linear of order 6
ρ1211-1-11ζ32ζ3ζ3ζ321ζ32ζ3ζ3ζ32ζ6ζ65ζ65ζ61111-1-1-1-11111    linear of order 6
ρ132200-122-1-1-122-1-1000022220000-1-1-1-1    orthogonal lifted from S3
ρ142-200-122-1-11-2-211000022-2-20000-1-111    orthogonal lifted from D6
ρ152-200-1-1+√-3-1-√-3ζ6ζ6511-√-31+√-3ζ32ζ3000022-2-20000-1-111    complex lifted from S3×C6
ρ162200-1-1-√-3-1+√-3ζ65ζ6-1-1-√-3-1+√-3ζ65ζ6000022220000-1-1-1-1    complex lifted from C3×S3
ρ172200-1-1+√-3-1-√-3ζ6ζ65-1-1+√-3-1-√-3ζ6ζ65000022220000-1-1-1-1    complex lifted from C3×S3
ρ182-200-1-1-√-3-1+√-3ζ65ζ611+√-31-√-3ζ3ζ32000022-2-20000-1-111    complex lifted from S3×C6
ρ19333330000300000000-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2    complex lifted from C7⋊C3
ρ2033-3-330000300000000-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/21+√-7/21-√-7/21-√-7/21+√-7/2-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2    complex lifted from C2×C7⋊C3
ρ213-3-3330000-300000000-1+√-7/2-1-√-7/21-√-7/21+√-7/21-√-7/2-1-√-7/21+√-7/2-1+√-7/2-1+√-7/2-1-√-7/21+√-7/21-√-7/2    complex lifted from C2×C7⋊C3
ρ223-33-330000-300000000-1+√-7/2-1-√-7/21-√-7/21+√-7/2-1+√-7/21+√-7/2-1-√-7/21-√-7/2-1+√-7/2-1-√-7/21+√-7/21-√-7/2    complex lifted from C2×C7⋊C3
ρ2333-3-330000300000000-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/21-√-7/21+√-7/21+√-7/21-√-7/2-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2    complex lifted from C2×C7⋊C3
ρ24333330000300000000-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2    complex lifted from C7⋊C3
ρ253-33-330000-300000000-1-√-7/2-1+√-7/21+√-7/21-√-7/2-1-√-7/21-√-7/2-1+√-7/21+√-7/2-1-√-7/2-1+√-7/21-√-7/21+√-7/2    complex lifted from C2×C7⋊C3
ρ263-3-3330000-300000000-1-√-7/2-1+√-7/21+√-7/21-√-7/21+√-7/2-1+√-7/21-√-7/2-1-√-7/2-1-√-7/2-1+√-7/21-√-7/21+√-7/2    complex lifted from C2×C7⋊C3
ρ276600-30000-300000000-1+√-7-1-√-7-1+√-7-1-√-700001-√-7/21+√-7/21+√-7/21-√-7/2    complex lifted from S3×C7⋊C3
ρ286-600-30000300000000-1-√-7-1+√-71+√-71-√-700001+√-7/21-√-7/2-1+√-7/2-1-√-7/2    complex faithful
ρ296600-30000-300000000-1-√-7-1+√-7-1-√-7-1+√-700001+√-7/21-√-7/21-√-7/21+√-7/2    complex lifted from S3×C7⋊C3
ρ306-600-30000300000000-1+√-7-1-√-71-√-71+√-700001-√-7/21+√-7/2-1-√-7/2-1+√-7/2    complex faithful

Smallest permutation representation of C2×S3×C7⋊C3
►On 42 points
Generators in S42
(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 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)

(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(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)

(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)| (1,22)(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), (1,8,15)(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), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(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), (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( (1,22)(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), (1,8,15)(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), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(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), (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([[(1,22),(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)], [(1,8,15),(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)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(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)], [(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)]])
 

Matrix representation of C2×S3×C7⋊C3 ►in GL5(𝔽43)

420000
042000
00100
00010
00001
,
16000
2141000
00100
00010
00001
,
4237000
01000
00100
00010
00001
,
10000
01000
0024251
00100
00010
,
60000
06000
00100
00184242
00010

G:=sub<GL(5,GF(43))| [42,0,0,0,0,0,42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,21,0,0,0,6,41,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[42,0,0,0,0,37,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,24,1,0,0,0,25,0,1,0,0,1,0,0],[6,0,0,0,0,0,6,0,0,0,0,0,1,18,0,0,0,0,42,1,0,0,0,42,0] >;
 

C2×S3×C7⋊C3 in GAP, Magma, Sage, TeX 

C_2\times S_3\times C_7\rtimes C_3
 
% in TeX
 
G:=Group("C2xS3xC7:C3");
 
// GroupNames label
 
G:=SmallGroup(252,29);
 
// by ID
 
G=gap.SmallGroup(252,29);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-7,483,464]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=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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Subgroup lattice of C2×S3×C7⋊C3 in TeX
Character table of C2×S3×C7⋊C3 in TeX

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