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G = C3×C12⋊7D4  order 288 = 25·32

Direct product of C3 and C12⋊7D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12⋊7D4, C62⋊18D4, C62.198C23, D6⋊C4⋊3C6, C12⋊7(C3×D4), (C2×C6)⋊7D12, (C2×D12)⋊5C6, (C3×C12)⋊23D4, C4⋊Dic3⋊9C6, C6.43(C6×D4), (C6×D12)⋊29C2, C2.17(C6×D12), C22⋊3(C3×D12), C12⋊14(C3⋊D4), (C22×C12)⋊15S3, (C22×C12)⋊10C6, (C2×C12).448D6, C6.105(C2×D12), C23.34(S3×C6), C32⋊20(C4⋊D4), (C22×C6).129D6, C6.128(C4○D12), (C6×C12).284C22, (C2×C62).101C22, (C6×Dic3).99C22, (C2×C6×C12)⋊12C2, (C2×C6)⋊8(C3×D4), C4⋊3(C3×C3⋊D4), C3⋊3(C3×C4⋊D4), (C2×C3⋊D4)⋊3C6, C2.7(C6×C3⋊D4), (C3×D6⋊C4)⋊34C2, (C6×C3⋊D4)⋊17C2, (C2×C4).85(S3×C6), (C22×C4)⋊8(C3×S3), C6.18(C3×C4○D4), C22.56(S3×C2×C6), (C2×C12).93(C2×C6), (C3×C4⋊Dic3)⋊33C2, C2.19(C3×C4○D12), (C3×C6).187(C2×D4), C6.144(C2×C3⋊D4), (S3×C2×C6).60C22, (C22×C6).65(C2×C6), (C2×C6).53(C22×C6), (C3×C6).106(C4○D4), (C22×S3).10(C2×C6), (C2×C6).331(C22×S3), (C2×Dic3).11(C2×C6), SmallGroup(288,701)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C3×C12⋊7D4
Chief series
C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C3×C12⋊7D4
Lower central
C3 — C2×C6 — C3×C12⋊7D4
Upper central
C1 — C2×C6 — C22×C12

Generators and relations for C3×C12⋊7D4
 G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 538 in 215 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C62, C62, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, C6×C12, S3×C2×C6, C2×C62, C12⋊7D4, C3×C4⋊D4, C3×C4⋊Dic3, C3×D6⋊C4, C6×D12, C6×C3⋊D4, C2×C6×C12, C3×C12⋊7D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, D12, C3⋊D4, C3×D4, C22×S3, C22×C6, C4⋊D4, S3×C6, C2×D12, C4○D12, C2×C3⋊D4, C6×D4, C3×C4○D4, C3×D12, C3×C3⋊D4, S3×C2×C6, C12⋊7D4, C3×C4⋊D4, C6×D12, C3×C4○D12, C6×C3⋊D4, C3×C12⋊7D4

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6AE6AF6AG6AH6AI12A···12AF12AG12AH12AI12AJ
order12222222333334444446···66···6666612···1212121212
size111122121211222222212121···12···2121212122···212121212

90 irreducible representations

dim111111111111222222222222222222
type++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C4○D4C3×S3C3⋊D4C3×D4D12C3×D4S3×C6S3×C6C4○D12C3×C4○D4C3×C3⋊D4C3×D12C3×C4○D12
kernelC3×C12⋊7D4C3×C4⋊Dic3C3×D6⋊C4C6×D12C6×C3⋊D4C2×C6×C12C12⋊7D4C4⋊Dic3D6⋊C4C2×D12C2×C3⋊D4C22×C12C22×C12C3×C12C62C2×C12C22×C6C3×C6C22×C4C12C12C2×C6C2×C6C2×C4C23C6C6C4C22C2
# reps112121224242122212244444244888

Matrix representation of C3×C12⋊7D4 ►in GL4(𝔽13) generated by

3000
0300
0030
0003
,
4400
01000
0070
00112
,
121000
5100
00711
00116
,
1000
81200
0062
0027
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,4,10,0,0,0,0,7,11,0,0,0,2],[12,5,0,0,10,1,0,0,0,0,7,11,0,0,11,6],[1,8,0,0,0,12,0,0,0,0,6,2,0,0,2,7] >;
 

C3×C12⋊7D4 in GAP, Magma, Sage, TeX 

C_3\times C_{12}\rtimes_7D_4
 
% in TeX
 
G:=Group("C3xC12:7D4");
 
// GroupNames label
 
G:=SmallGroup(288,701);
 
// by ID
 
G=gap.SmallGroup(288,701);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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