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G = C3×S3×C4○D4  order 288 = 25·32

Direct product of C3, S3 and C4○D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×S3×C4○D4, C62.150C23, (S3×D4)⋊6C6, D4⋊7(S3×C6), (S3×Q8)⋊9C6, Q8⋊8(S3×C6), C4○D12⋊7C6, (C3×D4)⋊25D6, (C2×C12)⋊25D6, (C3×Q8)⋊25D6, D4⋊2S3⋊6C6, D12⋊10(C2×C6), Q8⋊3S3⋊9C6, (C6×C12)⋊11C22, Dic6⋊10(C2×C6), (C3×C6).48C24, C6.79(S3×C23), C6.11(C23×C6), (S3×C12)⋊24C22, (C3×D12)⋊36C22, (S3×C6).33C23, C12.25(C22×C6), D6.10(C22×C6), C12.176(C22×S3), (C3×C12).126C23, (C3×Dic6)⋊35C22, (C6×Dic3)⋊35C22, (D4×C32)⋊21C22, (Q8×C32)⋊19C22, (C3×Dic3).34C23, Dic3.10(C22×C6), (S3×C2×C4)⋊6C6, (C2×C4)⋊7(S3×C6), (C3×S3×D4)⋊13C2, C3⋊4(C6×C4○D4), C4.25(S3×C2×C6), (C3×S3×Q8)⋊13C2, (S3×C2×C12)⋊14C2, (C4×S3)⋊7(C2×C6), (C2×C12)⋊4(C2×C6), (C3×C4○D4)⋊7C6, (C3×D4)⋊8(C2×C6), C3⋊D4⋊4(C2×C6), (C3×Q8)⋊9(C2×C6), C22.2(S3×C2×C6), C32⋊19(C2×C4○D4), C2.12(S3×C22×C6), (C3×C4○D12)⋊17C2, (C32×C4○D4)⋊5C2, (C2×C6).3(C22×C6), (C3×D4⋊2S3)⋊13C2, (C3×Q8⋊3S3)⋊13C2, (C2×Dic3)⋊10(C2×C6), (C3×C3⋊D4)⋊17C22, (S3×C2×C6).112C22, (C2×C6).22(C22×S3), (C22×S3).32(C2×C6), SmallGroup(288,998)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C3×S3×C4○D4
Chief series
C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C2×C12 — C3×S3×C4○D4
Lower central
C3 — C6 — C3×S3×C4○D4
Upper central
C1 — C12 — C3×C4○D4

Generators and relations for C3×S3×C4○D4
 G = < a,b,c,d,e,f | a3=b3=c2=d4=f2=1, e2=d2, 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, de=ed, df=fd, fef=d2e >

Subgroups: 706 in 348 conjugacy classes, 174 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C32, Dic3, Dic3, C12, C12, C12, D6, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3×S3, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C2×C4○D4, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, S3×C6, C62, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C3×Dic6, S3×C12, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, Q8×C32, S3×C2×C6, S3×C4○D4, C6×C4○D4, S3×C2×C12, C3×C4○D12, C3×S3×D4, C3×D4⋊2S3, C3×S3×Q8, C3×Q8⋊3S3, C32×C4○D4, C3×S3×C4○D4
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C24, C3×S3, C22×S3, C22×C6, C2×C4○D4, S3×C6, C3×C4○D4, S3×C23, C23×C6, S3×C2×C6, S3×C4○D4, C6×C4○D4, S3×C22×C6, C3×S3×C4○D4

Smallest permutation representation of C3×S3×C4○D4
►On 48 points
Generators in S48
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 25)(10 24 26)(11 21 27)(12 22 28)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(29 35 37)(30 36 38)(31 33 39)(32 34 40)

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

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C3D3E4A4B4C4D4E4F4G4H4I4J6A6B6C···6K6L6M6N6O6P···6X6Y···6AD12A12B12C12D12E···12P12Q12R12S12T12U···12AC12AD···12AI
order1222222222333334444444444666···666666···66···61212121212···121212121212···1212···12
size1122233666112221122233666112···233334···46···611112···233334···46···6

90 irreducible representations

dim1111111111111111222222222244
type++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D6D6D6C4○D4C3×S3S3×C6S3×C6S3×C6C3×C4○D4S3×C4○D4C3×S3×C4○D4
kernelC3×S3×C4○D4S3×C2×C12C3×C4○D12C3×S3×D4C3×D4⋊2S3C3×S3×Q8C3×Q8⋊3S3C32×C4○D4S3×C4○D4S3×C2×C4C4○D12S3×D4D4⋊2S3S3×Q8Q8⋊3S3C3×C4○D4C3×C4○D4C2×C12C3×D4C3×Q8C3×S3C4○D4C2×C4D4Q8S3C3C1
# reps1333311126666222133142662824

Matrix representation of C3×S3×C4○D4 ►in GL4(𝔽13) generated by

9000
0900
0010
0001
,
9000
0300
0010
0001
,
0100
1000
00120
00012
,
1000
0100
0080
0008
,
1000
0100
0080
0035
,
1000
0100
00121
0001
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,8,3,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,1,1] >;
 

C3×S3×C4○D4 in GAP, Magma, Sage, TeX 

C_3\times S_3\times C_4\circ D_4
 
% in TeX
 
G:=Group("C3xS3xC4oD4");
 
// GroupNames label
 
G:=SmallGroup(288,998);
 
// by ID
 
G=gap.SmallGroup(288,998);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,794,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^4=f^2=1,e^2=d^2,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,d*e=e*d,d*f=f*d,f*e*f=d^2*e>;
 
// generators/relations
 

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