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G = C2×C6.D12  order 288 = 25·32

Direct product of C2 and C6.D12

direct product, metabelian, supersoluble, monomial

Aliases: C2×C6.D12, C62.58D4, C62.105C23, C6⋊1(D6⋊C4), C23.41S32, C6.83(C2×D12), (C2×C6).65D12, (C2×Dic3)⋊18D6, C62.55(C2×C4), (C22×Dic3)⋊7S3, (C22×C6).114D6, (C6×Dic3)⋊22C22, (C2×C62).24C22, C22.24(C3⋊D12), C22.15(C6.D6), C3⋊2(C2×D6⋊C4), C6.37(S3×C2×C4), (Dic3×C2×C6)⋊3C2, (C22×C3⋊S3)⋊4C4, (C2×C6).31(C4×S3), C22.51(C2×S32), C6.20(C2×C3⋊D4), C32⋊6(C2×C22⋊C4), (C3×C6)⋊4(C22⋊C4), C2.3(C2×C3⋊D12), (C3×C6).151(C2×D4), (C23×C3⋊S3).1C2, (C2×C6).41(C3⋊D4), (C3×C6).65(C22×C4), C2.14(C2×C6.D6), (C2×C6).124(C22×S3), (C22×C3⋊S3).76C22, (C2×C3⋊S3)⋊14(C2×C4), SmallGroup(288,611)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×C6.D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C6.D12 — C2×C6.D12
Lower central
C32 — C3×C6 — C2×C6.D12
Upper central
C1 — C23

Generators and relations for C2×C6.D12
 G = < a,b,c,d | a2=b6=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 1474 in 331 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C3×Dic3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C6×Dic3, C6×Dic3, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C2×D6⋊C4, C6.D12, Dic3×C2×C6, C23×C3⋊S3, C2×C6.D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, S32, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C6.D6, C3⋊D12, C2×S32, C2×D6⋊C4, C6.D12, C2×C6.D6, C2×C3⋊D12, C2×C6.D12

Smallest permutation representation of C2×C6.D12
►On 48 points
Generators in S48
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 37)(21 38)(22 39)(23 40)(24 41)

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

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

60 conjugacy classes

class 1 2A···2G2H2I2J2K3A3B3C4A···4H6A···6N6O···6U12A···12P
order12···222223334···46···66···612···12
size11···1181818182246···62···24···46···6

60 irreducible representations

dim1111122222224444
type+++++++++++++
imageC1C2C2C2C4S3D4D6D6C4×S3D12C3⋊D4S32C6.D6C3⋊D12C2×S32
kernelC2×C6.D12C6.D12Dic3×C2×C6C23×C3⋊S3C22×C3⋊S3C22×Dic3C62C2×Dic3C22×C6C2×C6C2×C6C2×C6C23C22C22C22
# reps1421824428881241

Matrix representation of C2×C6.D12 ►in GL6(𝔽13)

1200000
0120000
001000
000100
000010
000001
,
0120000
1120000
0012000
0001200
0000120
0000012
,
0120000
1200000
000100
001000
000005
000085
,
0120000
1200000
001000
0001200
000010
0000112

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

C2×C6.D12 in GAP, Magma, Sage, TeX 

C_2\times C_6.D_{12}
 
% in TeX
 
G:=Group("C2xC6.D12");
 
// GroupNames label
 
G:=SmallGroup(288,611);
 
// by ID
 
G=gap.SmallGroup(288,611);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,176,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^6=c^12=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=b^3*c^-1>;
 
// generators/relations
 

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