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G = C2xD6:D6  order 288 = 25·32

Direct product of C2 and D6:D6

direct product, metabelian, supersoluble, monomial

Aliases: C2xD6:D6, D12:22D6, C62.136C23, C6:2(S3xD4), (C2xC12):3D6, (C6xD12):16C2, (C2xD12):13S3, (S3xC6):2C23, C12:2(C22xS3), (C6xC12):4C22, (C3xC12):3C23, D6:2(C22xS3), C32:3(C22xD4), (C22xS3):10D6, (C3xC6).11C24, C6.11(S3xC23), C3:Dic3:4C23, (C3xD12):28C22, D6:S3:11C22, C4:3(C2xS32), (C2xC4):9S32, C3:2(C2xS3xD4), C3:S3:2(C2xD4), (C3xC6):3(C2xD4), (C2xC3:S3):14D4, (C22xS32):6C2, (C2xS32):9C22, (S3xC2xC6):7C22, C2.13(C22xS32), C22.66(C2xS32), (C4xC3:S3):12C22, (C2xD6:S3):14C2, (C2xC3:S3).44C23, (C2xC6).153(C22xS3), (C2xC3:Dic3):21C22, (C22xC3:S3).103C22, (C2xC4xC3:S3):5C2, SmallGroup(288,952)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C2xD6:D6
Chief series
C1 — C3 — C32 — C3xC6 — S3xC6 — C2xS32 — C22xS32 — C2xD6:D6
Lower central
C32 — C3xC6 — C2xD6:D6
Upper central
C1 — C22 — C2xC4

Generators and relations for C2xD6: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, C2xC4, C2xC4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C22xC4, C2xD4, C24, C3xS3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22xD4, C3:Dic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C62, S3xC2xC4, C2xD12, S3xD4, C2xC3:D4, C6xD4, S3xC23, D6:S3, C3xD12, C4xC3:S3, C2xC3:Dic3, C6xC12, C2xS32, C2xS32, S3xC2xC6, C22xC3:S3, C2xS3xD4, D6:D6, C2xD6:S3, C6xD12, C2xC4xC3:S3, C22xS32, C2xD6:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, S32, S3xD4, S3xC23, C2xS32, C2xS3xD4, D6:D6, C22xS32, C2xD6:D6

Smallest permutation representation of C2xD6: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+++++++++++++++
imageC1C2C2C2C2C2S3D4D6D6D6S32S3xD4C2xS32C2xS32D6:D6
kernelC2xD6:D6D6:D6C2xD6:S3C6xD12C2xC4xC3:S3C22xS32C2xD12C2xC3:S3D12C2xC12C22xS3C2xC4C6C4C22C2
# reps1822122482414214

Matrix representation of C2xD6: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] >;
 

C2xD6: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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