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G = C2×S3×C3⋊D4  order 288 = 25·32

Direct product of C2, S3 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C2×S3×C3⋊D4, C62⋊2C23, C23⋊4S32, C6⋊5(S3×D4), (S3×C6)⋊19D4, (S3×C23)⋊8S3, (S3×C6)⋊4C23, D6⋊7(C22×S3), (C22×C6)⋊13D6, C32⋊7(C22×D4), (C2×Dic3)⋊15D6, (C22×S3)⋊12D6, (C3×C6).35C24, C6.35(S3×C23), C3⋊Dic3⋊2C23, (C2×C62)⋊6C22, Dic3⋊2(C22×S3), (C6×Dic3)⋊8C22, (C3×Dic3)⋊2C23, C32⋊7D4⋊9C22, D6⋊S3⋊18C22, C3⋊D12⋊17C22, (S3×Dic3)⋊10C22, C3⋊6(C2×S3×D4), C22⋊3(C2×S32), (C3×C6)⋊6(C2×D4), C6⋊2(C2×C3⋊D4), (C22×S32)⋊8C2, (C3×S3)⋊3(C2×D4), (C6×C3⋊D4)⋊6C2, (S3×C22×C6)⋊7C2, (C2×S3×Dic3)⋊6C2, (C2×S32)⋊11C22, (C2×C3⋊S3)⋊3C23, (S3×C2×C6)⋊18C22, (C2×C6)⋊8(C22×S3), C2.35(C22×S32), C3⋊2(C22×C3⋊D4), (C2×C3⋊D12)⋊21C2, (C2×D6⋊S3)⋊17C2, (C3×C3⋊D4)⋊12C22, (C2×C32⋊7D4)⋊16C2, (C22×C3⋊S3)⋊9C22, (C2×C3⋊Dic3)⋊12C22, SmallGroup(288,976)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×S3×C3⋊D4
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C22×S32 — C2×S3×C3⋊D4
Lower central
C32 — C3×C6 — C2×S3×C3⋊D4
Upper central
C1 — C22 — C23

Generators and relations for C2×S3×C3⋊D4
 G = < a,b,c,d,e,f | a2=b3=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1922 in 499 conjugacy classes, 132 normal (36 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C22×D4, C3×Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, S3×C23, C23×C6, S3×Dic3, D6⋊S3, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C2×S32, C2×S32, S3×C2×C6, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, C2×C62, C2×S3×D4, C22×C3⋊D4, C2×S3×Dic3, C2×D6⋊S3, C2×C3⋊D12, S3×C3⋊D4, C6×C3⋊D4, C2×C32⋊7D4, C22×S32, S3×C22×C6, C2×S3×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, S32, S3×D4, C2×C3⋊D4, S3×C23, C2×S32, C2×S3×D4, C22×C3⋊D4, S3×C3⋊D4, C22×S32, C2×S3×C3⋊D4

Smallest permutation representation of C2×S3×C3⋊D4
►On 48 points
Generators in S48
(1 30)(2 31)(3 32)(4 29)(5 35)(6 36)(7 33)(8 34)(9 42)(10 43)(11 44)(12 41)(13 24)(14 21)(15 22)(16 23)(17 27)(18 28)(19 25)(20 26)(37 47)(38 48)(39 45)(40 46)

(1 44 45)(2 41 46)(3 42 47)(4 43 48)(5 22 26)(6 23 27)(7 24 28)(8 21 25)(9 37 32)(10 38 29)(11 39 30)(12 40 31)(13 18 33)(14 19 34)(15 20 35)(16 17 36)

(1 6)(2 7)(3 8)(4 5)(9 19)(10 20)(11 17)(12 18)(13 40)(14 37)(15 38)(16 39)(21 47)(22 48)(23 45)(24 46)(25 42)(26 43)(27 44)(28 41)(29 35)(30 36)(31 33)(32 34)

(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 26 22)(6 23 27)(7 28 24)(8 21 25)(9 32 37)(10 38 29)(11 30 39)(12 40 31)(13 33 18)(14 19 34)(15 35 20)(16 17 36)

(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)

(1 34)(2 33)(3 36)(4 35)(5 29)(6 32)(7 31)(8 30)(9 23)(10 22)(11 21)(12 24)(13 41)(14 44)(15 43)(16 42)(17 47)(18 46)(19 45)(20 48)(25 39)(26 38)(27 37)(28 40)
 
G:=sub<Sym(48)| (1,30)(2,31)(3,32)(4,29)(5,35)(6,36)(7,33)(8,34)(9,42)(10,43)(11,44)(12,41)(13,24)(14,21)(15,22)(16,23)(17,27)(18,28)(19,25)(20,26)(37,47)(38,48)(39,45)(40,46), (1,44,45)(2,41,46)(3,42,47)(4,43,48)(5,22,26)(6,23,27)(7,24,28)(8,21,25)(9,37,32)(10,38,29)(11,39,30)(12,40,31)(13,18,33)(14,19,34)(15,20,35)(16,17,36), (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,40)(14,37)(15,38)(16,39)(21,47)(22,48)(23,45)(24,46)(25,42)(26,43)(27,44)(28,41)(29,35)(30,36)(31,33)(32,34), (1,45,44)(2,41,46)(3,47,42)(4,43,48)(5,26,22)(6,23,27)(7,28,24)(8,21,25)(9,32,37)(10,38,29)(11,30,39)(12,40,31)(13,33,18)(14,19,34)(15,35,20)(16,17,36), (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), (1,34)(2,33)(3,36)(4,35)(5,29)(6,32)(7,31)(8,30)(9,23)(10,22)(11,21)(12,24)(13,41)(14,44)(15,43)(16,42)(17,47)(18,46)(19,45)(20,48)(25,39)(26,38)(27,37)(28,40)>;
 
G:=Group( (1,30)(2,31)(3,32)(4,29)(5,35)(6,36)(7,33)(8,34)(9,42)(10,43)(11,44)(12,41)(13,24)(14,21)(15,22)(16,23)(17,27)(18,28)(19,25)(20,26)(37,47)(38,48)(39,45)(40,46), (1,44,45)(2,41,46)(3,42,47)(4,43,48)(5,22,26)(6,23,27)(7,24,28)(8,21,25)(9,37,32)(10,38,29)(11,39,30)(12,40,31)(13,18,33)(14,19,34)(15,20,35)(16,17,36), (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,40)(14,37)(15,38)(16,39)(21,47)(22,48)(23,45)(24,46)(25,42)(26,43)(27,44)(28,41)(29,35)(30,36)(31,33)(32,34), (1,45,44)(2,41,46)(3,47,42)(4,43,48)(5,26,22)(6,23,27)(7,28,24)(8,21,25)(9,32,37)(10,38,29)(11,30,39)(12,40,31)(13,33,18)(14,19,34)(15,35,20)(16,17,36), (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), (1,34)(2,33)(3,36)(4,35)(5,29)(6,32)(7,31)(8,30)(9,23)(10,22)(11,21)(12,24)(13,41)(14,44)(15,43)(16,42)(17,47)(18,46)(19,45)(20,48)(25,39)(26,38)(27,37)(28,40) );
 
G=PermutationGroup([[(1,30),(2,31),(3,32),(4,29),(5,35),(6,36),(7,33),(8,34),(9,42),(10,43),(11,44),(12,41),(13,24),(14,21),(15,22),(16,23),(17,27),(18,28),(19,25),(20,26),(37,47),(38,48),(39,45),(40,46)], [(1,44,45),(2,41,46),(3,42,47),(4,43,48),(5,22,26),(6,23,27),(7,24,28),(8,21,25),(9,37,32),(10,38,29),(11,39,30),(12,40,31),(13,18,33),(14,19,34),(15,20,35),(16,17,36)], [(1,6),(2,7),(3,8),(4,5),(9,19),(10,20),(11,17),(12,18),(13,40),(14,37),(15,38),(16,39),(21,47),(22,48),(23,45),(24,46),(25,42),(26,43),(27,44),(28,41),(29,35),(30,36),(31,33),(32,34)], [(1,45,44),(2,41,46),(3,47,42),(4,43,48),(5,26,22),(6,23,27),(7,28,24),(8,21,25),(9,32,37),(10,38,29),(11,30,39),(12,40,31),(13,33,18),(14,19,34),(15,35,20),(16,17,36)], [(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)], [(1,34),(2,33),(3,36),(4,35),(5,29),(6,32),(7,31),(8,30),(9,23),(10,22),(11,21),(12,24),(13,41),(14,44),(15,43),(16,42),(17,47),(18,46),(19,45),(20,48),(25,39),(26,38),(27,37),(28,40)]])
 

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O3A3B3C4A4B4C4D6A···6J6K···6S6T···6AA6AB6AC12A12B
order122222222222222233344446···66···66···6661212
size1111223333666618182246618182···24···46···612121212

54 irreducible representations

dim111111111222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3S3D4D6D6D6D6C3⋊D4S32S3×D4C2×S32S3×C3⋊D4
kernelC2×S3×C3⋊D4C2×S3×Dic3C2×D6⋊S3C2×C3⋊D12S3×C3⋊D4C6×C3⋊D4C2×C32⋊7D4C22×S32S3×C22×C6C2×C3⋊D4S3×C23S3×C6C2×Dic3C3⋊D4C22×S3C22×C6D6C23C6C22C2
# reps111181111114147281234

Matrix representation of C2×S3×C3⋊D4 ►in GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1000
00000-100
00000010
00000001
,
-11000000
-10000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00100000
00010000
00001000
00000100
000000-11
000000-10
,
-10000000
0-1000000
000-10000
00100000
00000100
0000-1000
0000000-1
000000-10
,
10000000
01000000
00-100000
00010000
0000-1000
00000100
0000000-1
000000-10

G:=sub<GL(8,Integers())| [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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0] >;
 

C2×S3×C3⋊D4 in GAP, Magma, Sage, TeX 

C_2\times S_3\times C_3\rtimes D_4
 
% in TeX
 
G:=Group("C2xS3xC3:D4");
 
// GroupNames label
 
G:=SmallGroup(288,976);
 
// by ID
 
G=gap.SmallGroup(288,976);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,346,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^2=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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