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
C32 — C2×C12.29D6 |
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
(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 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 8A | ··· | 8P | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 4 | 4 | 4 | 3 | ··· | 3 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
72 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D6 | D6 | C4×S3 | C4×S3 | S3×C8 | S32 | C6.D6 | C2×S32 | C6.D6 | C12.29D6 |
kernel | C2×C12.29D6 | C12.29D6 | C6×C3⋊C8 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C2×C3⋊Dic3 | C22×C3⋊S3 | C2×C3⋊S3 | C2×C3⋊C8 | C3⋊C8 | C2×C12 | C12 | C2×C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 |
# reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 16 | 2 | 4 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 4 |
Matrix representation of C2×C12.29D6 ►in GL6(𝔽73)
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 | 72 |
0 | 0 | 0 | 0 | 1 | 72 |
0 | 22 | 0 | 0 | 0 | 0 |
51 | 22 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 |
0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
27 | 46 | 0 | 0 | 0 | 0 |
0 | 46 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 72 | 72 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
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