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G = C22×C6.D6  order 288 = 25·32

Direct product of C22 and C6.D6

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C22×C6.D6, C62.144C23, C23.45S32, C62⋊16(C2×C4), C32⋊4(C23×C4), (C2×Dic3)⋊24D6, C6.31(S3×C23), (C3×C6).31C24, Dic3⋊7(C22×S3), (C3×Dic3)⋊8C23, (C22×C6).121D6, (C22×Dic3)⋊13S3, (C6×Dic3)⋊32C22, (C2×C62).79C22, C6⋊2(S3×C2×C4), C3⋊2(S3×C22×C4), (C2×C6)⋊12(C4×S3), C2.3(C22×S32), C3⋊S3⋊2(C22×C4), (C22×C3⋊S3)⋊9C4, (C3×C6)⋊4(C22×C4), C22.68(C2×S32), (Dic3×C2×C6)⋊17C2, (C23×C3⋊S3).6C2, (C2×C3⋊S3).49C23, (C2×C6).159(C22×S3), (C22×C3⋊S3).105C22, (C2×C3⋊S3)⋊19(C2×C4), SmallGroup(288,972)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C22×C6.D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×Dic3 — C6.D6 — C2×C6.D6 — C22×C6.D6
Lower central
C32 — C22×C6.D6
Upper central
C1 — C23

Generators and relations for C22×C6.D6
 G = < a,b,c,d,e | a2=b2=c6=e2=1, d6=c3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d5 >

Subgroups: 1858 in 539 conjugacy classes, 188 normal (8 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C24, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C22×C6, C23×C4, C3×Dic3, C2×C3⋊S3, C62, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C6.D6, C6×Dic3, C22×C3⋊S3, C2×C62, S3×C22×C4, C2×C6.D6, Dic3×C2×C6, C23×C3⋊S3, C22×C6.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, S32, S3×C2×C4, S3×C23, C6.D6, C2×S32, S3×C22×C4, C2×C6.D6, C22×S32, C22×C6.D6

Smallest permutation representation of C22×C6.D6
►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 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)

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

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

72 conjugacy classes

class 1 2A···2G2H···2O3A3B3C4A···4P6A···6N6O···6U12A···12P
order12···22···23334···46···66···612···12
size11···19···92243···32···24···46···6

72 irreducible representations

dim111112222444
type++++++++++
imageC1C2C2C2C4S3D6D6C4×S3S32C6.D6C2×S32
kernelC22×C6.D6C2×C6.D6Dic3×C2×C6C23×C3⋊S3C22×C3⋊S3C22×Dic3C2×Dic3C22×C6C2×C6C23C22C22
# reps1122116212216143

Matrix representation of C22×C6.D6 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012000
000001200
00000010
00000001
,
10000000
01000000
001200000
000120000
000012000
000001200
00000010
00000001
,
012000000
11000000
00010000
0012120000
000012100
000012000
00000010
00000001
,
08000000
80000000
00010000
00100000
000001200
000012000
000000012
000000112
,
012000000
120000000
00010000
00100000
000001200
000012000
000000112
000000012

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

C22×C6.D6 in GAP, Magma, Sage, TeX 

C_2^2\times C_6.D_6
 
% in TeX
 
G:=Group("C2^2xC6.D6");
 
// GroupNames label
 
G:=SmallGroup(288,972);
 
// by ID
 
G=gap.SmallGroup(288,972);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^6=e^2=1,d^6=c^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
 
// generators/relations
 

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