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G = C2×C12.29D6  order 288 = 25·32

Direct product of C2 and C12.29D6

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

Aliases: C2×C12.29D6, C6⋊1(S3×C8), C3⋊C8⋊30D6, C12.46(C4×S3), C32⋊5(C22×C8), (C2×C12).300D6, C62.46(C2×C4), (C3×C12).145C23, (C6×C12).205C22, C12.144(C22×S3), C4.15(C6.D6), C22.12(C6.D6), C3⋊2(S3×C2×C8), (C6×C3⋊C8)⋊17C2, (C2×C3⋊S3)⋊6C8, (C2×C3⋊C8)⋊13S3, C3⋊S3⋊3(C2×C8), C4.91(C2×S32), (C3×C6)⋊4(C2×C8), C6.25(S3×C2×C4), (C2×C4).133S32, (C4×C3⋊S3).12C4, (C3×C3⋊C8)⋊35C22, (C2×C6).29(C4×S3), (C3×C12).86(C2×C4), (C22×C3⋊S3).9C4, C2.1(C2×C6.D6), (C4×C3⋊S3).88C22, (C2×C3⋊Dic3).18C4, C3⋊Dic3.40(C2×C4), (C3×C6).41(C22×C4), (C2×C4×C3⋊S3).15C2, (C2×C3⋊S3).34(C2×C4), SmallGroup(288,464)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C12.29D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C3⋊C8 — C12.29D6 — C2×C12.29D6
Lower central
C32 — C2×C12.29D6
Upper central
C1 — C2×C4

Generators and relations for C2×C12.29D6
 G = < a,b,c,d | a2=b12=1, c6=b3, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c5 >

Subgroups: 594 in 179 conjugacy classes, 68 normal (16 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C22×C4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C3×C3⋊C8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C2×C8, C12.29D6, C6×C3⋊C8, C2×C4×C3⋊S3, C2×C12.29D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, C22×C8, S32, S3×C8, S3×C2×C4, C6.D6, C2×S32, S3×C2×C8, C12.29D6, C2×C6.D6, C2×C12.29D6

Smallest permutation representation of C2×C12.29D6
►On 48 points
Generators in S48
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I8A···8P12A···12H12I12J12K12L24A···24P
order12222222333444444446···66668···812···121212121224···24
size11119999224111199992···24443···32···244446···6

72 irreducible representations

dim1111111122222244444
type+++++++++++
imageC1C2C2C2C4C4C4C8S3D6D6C4×S3C4×S3S3×C8S32C6.D6C2×S32C6.D6C12.29D6
kernelC2×C12.29D6C12.29D6C6×C3⋊C8C2×C4×C3⋊S3C4×C3⋊S3C2×C3⋊Dic3C22×C3⋊S3C2×C3⋊S3C2×C3⋊C8C3⋊C8C2×C12C12C2×C6C6C2×C4C4C4C22C2
# reps142142216242441611114

Matrix representation of C2×C12.29D6 ►in GL6(𝔽73)

7200000
0720000
0072000
0007200
000010
000001
,
4600000
0460000
001000
000100
0000072
0000172
,
0220000
51220000
001100
0072000
000001
000010
,
27460000
0460000
001000
00727200
000001
000010

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,51,0,0,0,0,22,22,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[27,0,0,0,0,0,46,46,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C2×C12.29D6 in GAP, Magma, Sage, TeX 

C_2\times C_{12}._{29}D_6
 
% in TeX
 
G:=Group("C2xC12.29D6");
 
// GroupNames label
 
G:=SmallGroup(288,464);
 
// by ID
 
G=gap.SmallGroup(288,464);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^12=1,c^6=b^3,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^5>;
 
// generators/relations
 

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