direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C3⋊S3⋊3C8, (C6×C12).9C4, C32⋊3(C22×C8), C62.13(C2×C4), C32⋊2C8⋊10C22, C3⋊Dic3.29C23, (C2×C3⋊S3)⋊7C8, C3⋊S3⋊4(C2×C8), (C3×C6)⋊2(C2×C8), (C4×C3⋊S3).15C4, C4.21(C2×C32⋊C4), (C3×C12).18(C2×C4), C2.1(C22×C32⋊C4), (C2×C32⋊2C8)⋊10C2, (C4×C3⋊S3).95C22, (C22×C3⋊S3).14C4, (C2×C4).11(C32⋊C4), C3⋊Dic3.50(C2×C4), (C3×C6).24(C22×C4), C22.15(C2×C32⋊C4), (C2×C3⋊Dic3).172C22, (C2×C4×C3⋊S3).25C2, (C2×C3⋊S3).44(C2×C4), SmallGroup(288,929)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8 — C2×C3⋊S3⋊3C8 |
C32 — C2×C3⋊S3⋊3C8 |
Generators and relations for C2×C3⋊S3⋊3C8
G = < a,b,c,d,e | a2=b3=c3=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc-1, dcd=c-1, ece-1=b-1c-1, de=ed >
Subgroups: 576 in 130 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C4 [×2], C4 [×2], C22, C22 [×6], S3 [×8], C6 [×6], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C2×C8 [×6], C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C22×C8, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×2], C32⋊2C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C3⋊S3⋊3C8 [×4], C2×C32⋊2C8 [×2], C2×C4×C3⋊S3, C2×C3⋊S3⋊3C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, C22×C8, C32⋊C4, C2×C32⋊C4 [×3], C3⋊S3⋊3C8 [×2], C22×C32⋊C4, C2×C3⋊S3⋊3C8
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)
(1 28 18)(2 29 19)(3 20 30)(4 21 31)(5 32 22)(6 25 23)(7 24 26)(8 17 27)(9 35 43)(10 36 44)(11 45 37)(12 46 38)(13 39 47)(14 40 48)(15 41 33)(16 42 34)
(2 19 29)(4 31 21)(6 23 25)(8 27 17)(10 44 36)(12 38 46)(14 48 40)(16 34 42)
(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 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)
G:=sub<Sym(48)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35), (1,28,18)(2,29,19)(3,20,30)(4,21,31)(5,32,22)(6,25,23)(7,24,26)(8,17,27)(9,35,43)(10,36,44)(11,45,37)(12,46,38)(13,39,47)(14,40,48)(15,41,33)(16,42,34), (2,19,29)(4,31,21)(6,23,25)(8,27,17)(10,44,36)(12,38,46)(14,48,40)(16,34,42), (17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35), (1,28,18)(2,29,19)(3,20,30)(4,21,31)(5,32,22)(6,25,23)(7,24,26)(8,17,27)(9,35,43)(10,36,44)(11,45,37)(12,46,38)(13,39,47)(14,40,48)(15,41,33)(16,42,34), (2,19,29)(4,31,21)(6,23,25)(8,27,17)(10,44,36)(12,38,46)(14,48,40)(16,34,42), (17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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) );
G=PermutationGroup([(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35)], [(1,28,18),(2,29,19),(3,20,30),(4,21,31),(5,32,22),(6,25,23),(7,24,26),(8,17,27),(9,35,43),(10,36,44),(11,45,37),(12,46,38),(13,39,47),(14,40,48),(15,41,33),(16,42,34)], [(2,19,29),(4,31,21),(6,23,25),(8,27,17),(10,44,36),(12,38,46),(14,48,40),(16,34,42)], [(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)])
48 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8P | 12A | ··· | 12H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 4 | 4 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 4 | ··· | 4 | 9 | ··· | 9 | 4 | ··· | 4 |
48 irreducible representations
Matrix representation of C2×C3⋊S3⋊3C8 ►in GL5(𝔽73)
72 | 0 | 0 | 0 | 0 |
0 | 72 | 0 | 0 | 0 |
0 | 0 | 72 | 0 | 0 |
0 | 0 | 0 | 72 | 0 |
0 | 0 | 0 | 0 | 72 |
1 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 0 | 0 |
0 | 1 | 72 | 0 | 0 |
0 | 0 | 0 | 72 | 1 |
0 | 0 | 0 | 72 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 72 |
0 | 0 | 0 | 1 | 72 |
72 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 |
72 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 27 | 0 | 0 |
0 | 27 | 0 | 0 | 0 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[72,0,0,0,0,0,0,0,0,27,0,0,0,27,0,0,1,0,0,0,0,0,1,0,0] >;
C2×C3⋊S3⋊3C8 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes S_3\rtimes_3C_8
% in TeX
G:=Group("C2xC3:S3:3C8");
// GroupNames label
G:=SmallGroup(288,929);
// by ID
G=gap.SmallGroup(288,929);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,100,80,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1,e*b*e^-1=b*c^-1,d*c*d=c^-1,e*c*e^-1=b^-1*c^-1,d*e=e*d>;
// generators/relations