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G = C3×D12⋊C4  order 288 = 25·32

Direct product of C3 and D12⋊C4

direct product, metabelian, supersoluble, monomial

Aliases: C3×D12⋊C4, D12⋊4C12, Dic6⋊4C12, C62.34D4, C32⋊10C4≀C2, (C3×D12)⋊6C4, C4.3(S3×C12), C12.51(C4×S3), C12.6(C2×C12), (C3×Dic6)⋊6C4, C4○D12.2C6, (C4×Dic3)⋊1C6, C12.63(C3×D4), (C2×C6).45D12, C6.51(D6⋊C4), (Dic3×C12)⋊6C2, (C3×C12).165D4, (C2×C12).316D6, (C3×M4(2))⋊8C6, M4(2)⋊4(C3×S3), (C3×M4(2))⋊8S3, C22.3(C3×D12), (C6×C12).46C22, C12.146(C3⋊D4), (C32×M4(2))⋊12C2, C3⋊2(C3×C4≀C2), (C2×C6).2(C3×D4), (C2×C4).37(S3×C6), C2.11(C3×D6⋊C4), C4.29(C3×C3⋊D4), (C2×C12).16(C2×C6), (C3×C12).42(C2×C4), (C3×C4○D12).4C2, C6.10(C3×C22⋊C4), (C3×C6).50(C22⋊C4), SmallGroup(288,259)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C3×D12⋊C4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4○D12 — C3×D12⋊C4
Lower central
C3 — C6 — C12 — C3×D12⋊C4
Upper central
C1 — C12 — C2×C12 — C3×M4(2)

Generators and relations for C3×D12⋊C4
 G = < a,b,c,d | a3=b12=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b7c >

Subgroups: 250 in 98 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C4≀C2, C3×Dic3, C3×C12, S3×C6, C62, C4×Dic3, C4×C12, C3×M4(2), C3×M4(2), C4○D12, C3×C4○D4, C3×C24, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D12⋊C4, C3×C4≀C2, Dic3×C12, C32×M4(2), C3×C4○D12, C3×D12⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C4≀C2, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, D12⋊C4, C3×C4≀C2, C3×D6⋊C4, C3×D12⋊C4

Smallest permutation representation of C3×D12⋊C4
►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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)

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

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F4G4H6A6B6C···6G6H6I6J6K6L8A8B12A12B12C12D12E···12L12M12N12O12P···12W12X12Y24A···24P
order12223333344444444666···666666881212121212···1212121212···12121224···24
size1121211222112666612112···244412124411112···24446···612124···4

72 irreducible representations

dim111111111111222222222222222244
type+++++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12S3D4D4D6C3×S3C4×S3C3⋊D4C3×D4D12C3×D4C4≀C2S3×C6S3×C12C3×C3⋊D4C3×D12C3×C4≀C2D12⋊C4C3×D12⋊C4
kernelC3×D12⋊C4Dic3×C12C32×M4(2)C3×C4○D12D12⋊C4C3×Dic6C3×D12C4×Dic3C3×M4(2)C4○D12Dic6D12C3×M4(2)C3×C12C62C2×C12M4(2)C12C12C12C2×C6C2×C6C32C2×C4C4C4C22C3C3C1
# reps111122222244111122222242444824

Matrix representation of C3×D12⋊C4 ►in GL4(𝔽73) generated by

8000
0800
0010
0001
,
9000
06500
004644
00027
,
06500
9000
004644
001027
,
04600
27000
002741
0001
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,65,0,0,0,0,46,0,0,0,44,27],[0,9,0,0,65,0,0,0,0,0,46,10,0,0,44,27],[0,27,0,0,46,0,0,0,0,0,27,0,0,0,41,1] >;
 

C3×D12⋊C4 in GAP, Magma, Sage, TeX 

C_3\times D_{12}\rtimes C_4
 
% in TeX
 
G:=Group("C3xD12:C4");
 
// GroupNames label
 
G:=SmallGroup(288,259);
 
// by ID
 
G=gap.SmallGroup(288,259);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,136,2524,1271,102,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*c>;
 
// generators/relations
 

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