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G = C2×D6⋊D6  order 288 = 25·32

Direct product of C2 and D6⋊D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×D6⋊D6, D12⋊22D6, C62.136C23, C6⋊2(S3×D4), (C2×C12)⋊3D6, (C6×D12)⋊16C2, (C2×D12)⋊13S3, (S3×C6)⋊2C23, C12⋊2(C22×S3), (C6×C12)⋊4C22, (C3×C12)⋊3C23, D6⋊2(C22×S3), C32⋊3(C22×D4), (C22×S3)⋊10D6, (C3×C6).11C24, C6.11(S3×C23), C3⋊Dic3⋊4C23, (C3×D12)⋊28C22, D6⋊S3⋊11C22, C4⋊3(C2×S32), (C2×C4)⋊9S32, C3⋊2(C2×S3×D4), C3⋊S3⋊2(C2×D4), (C3×C6)⋊3(C2×D4), (C2×C3⋊S3)⋊14D4, (C22×S32)⋊6C2, (C2×S32)⋊9C22, (S3×C2×C6)⋊7C22, C2.13(C22×S32), C22.66(C2×S32), (C4×C3⋊S3)⋊12C22, (C2×D6⋊S3)⋊14C2, (C2×C3⋊S3).44C23, (C2×C6).153(C22×S3), (C2×C3⋊Dic3)⋊21C22, (C22×C3⋊S3).103C22, (C2×C4×C3⋊S3)⋊5C2, SmallGroup(288,952)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×D6⋊D6
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C22×S32 — C2×D6⋊D6
Lower central
C32 — C3×C6 — C2×D6⋊D6
Upper central
C1 — C22 — C2×C4

Generators and relations for C2×D6⋊D6
 G = < a,b,c,d,e | a2=b6=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=b3c, ede=d-1 >

Subgroups: 2130 in 499 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×D4, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, D6⋊S3, C3×D12, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, C2×S3×D4, D6⋊D6, C2×D6⋊S3, C6×D12, C2×C4×C3⋊S3, C22×S32, C2×D6⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S32, S3×D4, S3×C23, C2×S32, C2×S3×D4, D6⋊D6, C22×S32, C2×D6⋊D6

Smallest permutation representation of C2×D6⋊D6
►On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 34)(8 33)(9 32)(10 31)(11 36)(12 35)(13 40)(14 39)(15 38)(16 37)(17 42)(18 41)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)

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

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

48 conjugacy classes

class 1 2A2B2C2D···2K2L2M2N2O3A3B3C4A4B4C4D6A···6F6G6H6I6J···6Q12A···12H
order12222···2222233344446···66666···612···12
size11116···699992242218182···244412···124···4

48 irreducible representations

dim1111112222244444
type+++++++++++++++
imageC1C2C2C2C2C2S3D4D6D6D6S32S3×D4C2×S32C2×S32D6⋊D6
kernelC2×D6⋊D6D6⋊D6C2×D6⋊S3C6×D12C2×C4×C3⋊S3C22×S32C2×D12C2×C3⋊S3D12C2×C12C22×S3C2×C4C6C4C22C2
# reps1822122482414214

Matrix representation of C2×D6⋊D6 ►in GL8(𝔽13)

10000000
01000000
001200000
000120000
00001000
00000100
00000010
00000001
,
10000000
01000000
001200000
000120000
000012000
000001200
000000012
000000112
,
10000000
01000000
00280000
0011110000
000071100
000011600
000000112
000000012
,
112000000
10000000
00370000
0010100000
000041000
00005900
00000001
00000010
,
10000000
112000000
001060000
00330000
00009300
00008400
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,11,0,0,0,0,0,0,8,11,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,11,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12],[1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,3,10,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,4,5,0,0,0,0,0,0,10,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,3,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,9,8,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C2×D6⋊D6 in GAP, Magma, Sage, TeX 

C_2\times D_6\rtimes D_6
 
% in TeX
 
G:=Group("C2xD6:D6");
 
// GroupNames label
 
G:=SmallGroup(288,952);
 
// by ID
 
G=gap.SmallGroup(288,952);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=b^3*c,e*d*e=d^-1>;
 
// generators/relations
 

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