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G = C3×C12.55D4  order 288 = 25·32

Direct product of C3 and C12.55D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12.55D4, C62⋊6C8, (C2×C6)⋊4C24, (C6×C12).17C4, C6.10(C2×C24), C12.64(C3×D4), (C2×C12).10C12, (C3×C12).166D4, (C2×C12).457D6, C32⋊9(C22⋊C8), (C2×C62).10C4, C62.95(C2×C4), C6.6(C3×M4(2)), (C22×C12).24C6, (C22×C6).14C12, (C22×C12).13S3, (C2×C12).10Dic3, (C3×C6).20M4(2), (C22×C6).9Dic3, C23.4(C3×Dic3), C12.147(C3⋊D4), (C6×C12).335C22, C6.13(C4.Dic3), C22.10(C6×Dic3), C6.29(C6.D4), (C6×C3⋊C8)⋊7C2, C2.5(C6×C3⋊C8), (C2×C3⋊C8)⋊10C6, (C2×C6)⋊3(C3⋊C8), C6.21(C2×C3⋊C8), C22⋊3(C3×C3⋊C8), C3⋊2(C3×C22⋊C8), (C2×C6×C12).18C2, (C3×C6).41(C2×C8), (C2×C4).94(S3×C6), C4.30(C3×C3⋊D4), (C2×C6).37(C2×C12), C6.12(C3×C22⋊C4), (C22×C4).5(C3×S3), (C2×C4).3(C3×Dic3), (C2×C12).124(C2×C6), C2.3(C3×C4.Dic3), (C2×C6).59(C2×Dic3), C2.1(C3×C6.D4), (C3×C6).62(C22⋊C4), SmallGroup(288,264)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C3×C12.55D4
Chief series
C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C6×C3⋊C8 — C3×C12.55D4
Lower central
C3 — C6 — C3×C12.55D4
Upper central
C1 — C2×C12 — C22×C12

Generators and relations for C3×C12.55D4
 G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b3c3 >

Subgroups: 218 in 127 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C32, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C22×C4, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22⋊C8, C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C2×C24, C22×C12, C22×C12, C3×C3⋊C8, C6×C12, C6×C12, C2×C62, C12.55D4, C3×C22⋊C8, C6×C3⋊C8, C2×C6×C12, C3×C12.55D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C2×C8, M4(2), C3×S3, C3⋊C8, C24, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22⋊C8, C3×Dic3, S3×C6, C2×C3⋊C8, C4.Dic3, C6.D4, C3×C22⋊C4, C2×C24, C3×M4(2), C3×C3⋊C8, C6×Dic3, C3×C3⋊D4, C12.55D4, C3×C22⋊C8, C6×C3⋊C8, C3×C4.Dic3, C3×C6.D4, C3×C12.55D4

Smallest permutation representation of C3×C12.55D4
►On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 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 42 14 30 7 48 20 36)(2 47 15 35 8 41 21 29)(3 40 16 28 9 46 22 34)(4 45 17 33 10 39 23 27)(5 38 18 26 11 44 24 32)(6 43 19 31 12 37 13 25)

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

108 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6AE8A···8H12A···12H12I···12AJ24A···24P
order122222333334444446···66···68···812···1212···1224···24
size111122112221111221···12···26···61···12···26···6

108 irreducible representations

dim111111111111222222222222222222
type+++++-+-
imageC1C2C2C3C4C4C6C6C8C12C12C24S3D4Dic3D6Dic3M4(2)C3×S3C3⋊D4C3×D4C3⋊C8C3×Dic3S3×C6C3×Dic3C4.Dic3C3×M4(2)C3×C3⋊D4C3×C3⋊C8C3×C4.Dic3
kernelC3×C12.55D4C6×C3⋊C8C2×C6×C12C12.55D4C6×C12C2×C62C2×C3⋊C8C22×C12C62C2×C12C22×C6C2×C6C22×C12C3×C12C2×C12C2×C12C22×C6C3×C6C22×C4C12C12C2×C6C2×C4C2×C4C23C6C6C4C22C2
# reps1212224284416121112244422244888

Matrix representation of C3×C12.55D4 ►in GL3(𝔽73) generated by

100
0640
0064
,
2700
030
02124
,
6300
06363
0010
,
1000
06363
02010
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[27,0,0,0,3,21,0,0,24],[63,0,0,0,63,0,0,63,10],[10,0,0,0,63,20,0,63,10] >;
 

C3×C12.55D4 in GAP, Magma, Sage, TeX 

C_3\times C_{12}._{55}D_4
 
% in TeX
 
G:=Group("C3xC12.55D4");
 
// GroupNames label
 
G:=SmallGroup(288,264);
 
// by ID
 
G=gap.SmallGroup(288,264);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,136,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
 
// generators/relations
 

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