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G = C2×C12.29D6order 288 = 25·32

Direct product of C2 and C12.29D6

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

Aliases: C2×C12.29D6, C61(S3×C8), C3⋊C830D6, C12.46(C4×S3), C325(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), C32(S3×C2×C8), (C6×C3⋊C8)⋊17C2, (C2×C3⋊S3)⋊6C8, (C2×C3⋊C8)⋊13S3, C3⋊S33(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

C1C32 — C2×C12.29D6
C1C3C32C3×C6C3×C12C3×C3⋊C8C12.29D6 — C2×C12.29D6
C32 — C2×C12.29D6
C1C2×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 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×6], C12 [×4], C12 [×2], D6 [×18], C2×C6 [×2], C2×C6, C2×C8 [×6], C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×2], C2×C12, C22×S3 [×3], C22×C8, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, S3×C8 [×8], C2×C3⋊C8 [×2], C2×C24 [×2], S3×C2×C4 [×3], C3×C3⋊C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C2×C8 [×2], C12.29D6 [×4], C6×C3⋊C8 [×2], C2×C4×C3⋊S3, C2×C12.29D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C8 [×4], C2×C4 [×6], C23, D6 [×6], C2×C8 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], C22×C8, S32, S3×C8 [×4], S3×C2×C4 [×2], C6.D6 [×2], C2×S32, S3×C2×C8 [×2], C12.29D6 [×2], C2×C6.D6, C2×C12.29D6

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

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

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

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

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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