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G = C3×Dic3⋊C4  order 144 = 24·32

Direct product of C3 and Dic3⋊C4

direct product, metabelian, supersoluble, monomial

Aliases: C3×Dic3⋊C4, Dic3⋊C12, C6.8Dic6, C62.17C22, C6.5(C3×D4), (C3×C6).4Q8, C6.1(C3×Q8), C6.24(C4×S3), C2.4(S3×C12), C32⋊6(C4⋊C4), (C6×C12).1C2, (C2×C12).3S3, C6.3(C2×C12), (C2×C12).3C6, (C3×C6).27D4, (C2×C6).42D6, (C3×Dic3)⋊3C4, C22.4(S3×C6), C2.1(C3×Dic6), C6.27(C3⋊D4), (C2×Dic3).1C6, (C6×Dic3).6C2, C3⋊1(C3×C4⋊C4), (C2×C6).7(C2×C6), (C2×C4).1(C3×S3), C2.1(C3×C3⋊D4), (C3×C6).20(C2×C4), SmallGroup(144,77)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C3×Dic3⋊C4
Chief series
C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — C3×Dic3⋊C4
Lower central
C3 — C6 — C3×Dic3⋊C4
Upper central
C1 — C2×C6 — C2×C12

Generators and relations for C3×Dic3⋊C4
 G = < a,b,c,d | a3=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 104 in 60 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×Dic3, C3×Dic3, C3×C12, C62, Dic3⋊C4, C3×C4⋊C4, C6×Dic3, C6×C12, C3×Dic3⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, S3×C6, Dic3⋊C4, C3×C4⋊C4, C3×Dic6, S3×C12, C3×C3⋊D4, C3×Dic3⋊C4

Smallest permutation representation of C3×Dic3⋊C4
►On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 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 32 4 35)(2 31 5 34)(3 36 6 33)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)

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

C3×Dic3⋊C4 is a maximal subgroup of
 C62.6C23  Dic3⋊5Dic6  C62.8C23  C62.9C23  C62.10C23  Dic3.Dic6  C62.16C23  C62.17C23  C62.18C23  C62.20C23  D6⋊Dic6  C62.23C23  C62.31C23  C62.32C23  C62.35C23  C62.37C23  C62.38C23  C62.40C23  C62.48C23  C62.51C23  C62.53C23  C62.54C23  Dic3⋊D12  C62.58C23  D6⋊2Dic6  C62.65C23  D6⋊3Dic6  C62.67C23  C62.74C23  Dic3⋊3D12  C12×Dic6  C3×S3×C4⋊C4  C12×C3⋊D4  C62.19D6  Dic9⋊C12  C62.29D6
C3×Dic3⋊C4 is a maximal quotient of
 C62.19D6  Dic9⋊C12

54 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6O12A···12P12Q···12X
order1222333334444446···66···612···1212···12
size1111112222266661···12···22···26···6

54 irreducible representations

dim1111111122222222222222
type+++++-+-
imageC1C2C2C3C4C6C6C12S3D4Q8D6C3×S3Dic6C4×S3C3⋊D4C3×D4C3×Q8S3×C6C3×Dic6S3×C12C3×C3⋊D4
kernelC3×Dic3⋊C4C6×Dic3C6×C12Dic3⋊C4C3×Dic3C2×Dic3C2×C12Dic3C2×C12C3×C6C3×C6C2×C6C2×C4C6C6C6C6C6C22C2C2C2
# reps1212442811112222222444

Matrix representation of C3×Dic3⋊C4 ►in GL5(𝔽13)

90000
01000
00100
00030
00003
,
10000
012000
001200
00030
00009
,
10000
06900
06700
00001
00010
,
80000
011100
001200
000120
000012

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,6,6,0,0,0,9,7,0,0,0,0,0,0,1,0,0,0,1,0],[8,0,0,0,0,0,1,0,0,0,0,11,12,0,0,0,0,0,12,0,0,0,0,0,12] >;
 

C3×Dic3⋊C4 in GAP, Magma, Sage, TeX 

C_3\times {\rm Dic}_3\rtimes C_4
 
% in TeX
 
G:=Group("C3xDic3:C4");
 
// GroupNames label
 
G:=SmallGroup(144,77);
 
// by ID
 
G=gap.SmallGroup(144,77);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,313,79,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
 
// generators/relations
 

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