metabelian, supersoluble, monomial
Aliases: D12.2Dic3, Dic6.2Dic3, C3⋊C8.31D6, C3⋊5(C8○D12), C12.30(C4×S3), C4○D12.6S3, (C4×S3).31D6, (C3×D12).1C4, C32⋊4(C8○D4), C4.4(S3×Dic3), (C2×C12).107D6, C62.44(C2×C4), (C3×Dic6).1C4, D6.1(C2×Dic3), C3⋊D4.2Dic3, C12.58D6⋊8C2, D6.Dic3⋊10C2, (S3×C12).8C22, C3⋊1(D4.Dic3), (C6×C12).66C22, C12.25(C2×Dic3), C22.1(S3×Dic3), C6.3(C22×Dic3), (C3×C12).143C23, C12.142(C22×S3), Dic3.1(C2×Dic3), C32⋊4C8.20C22, (C2×C3⋊C8)⋊3S3, (S3×C3⋊C8)⋊9C2, (C6×C3⋊C8)⋊16C2, C4.89(C2×S32), (C2×C4).62S32, C6.83(S3×C2×C4), C2.5(C2×S3×Dic3), (S3×C6).4(C2×C4), (C2×C6).16(C4×S3), (C3×C3⋊D4).1C4, (C3×C12).56(C2×C4), (C3×C4○D12).5C2, (C3×C3⋊C8).38C22, (C3×C6).39(C22×C4), (C3×Dic3).4(C2×C4), (C2×C6).18(C2×Dic3), SmallGroup(288,462)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.2Dic3
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a6c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 338 in 134 conjugacy classes, 60 normal (38 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×6], C8 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×6], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C2×C3⋊C8 [×2], C4.Dic3 [×5], C2×C24, C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C32⋊4C8 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, C8○D12, D4.Dic3, S3×C3⋊C8 [×2], D6.Dic3 [×2], C6×C3⋊C8, C12.58D6, C3×C4○D12, D12.2Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C8○D4, S32, S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, C8○D12, D4.Dic3, C2×S3×Dic3, D12.2Dic3
(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 36)(12 35)(13 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 18 23 16 21 14 19 24 17 22 15 20)(25 36 35 34 33 32 31 30 29 28 27 26)(37 44 39 46 41 48 43 38 45 40 47 42)
(1 22 10 19 7 16 4 13)(2 23 11 20 8 17 5 14)(3 24 12 21 9 18 6 15)(25 44 28 47 31 38 34 41)(26 45 29 48 32 39 35 42)(27 46 30 37 33 40 36 43)
G:=sub<Sym(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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,36)(12,35)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39), (1,2,3,4,5,6,7,8,9,10,11,12)(13,18,23,16,21,14,19,24,17,22,15,20)(25,36,35,34,33,32,31,30,29,28,27,26)(37,44,39,46,41,48,43,38,45,40,47,42), (1,22,10,19,7,16,4,13)(2,23,11,20,8,17,5,14)(3,24,12,21,9,18,6,15)(25,44,28,47,31,38,34,41)(26,45,29,48,32,39,35,42)(27,46,30,37,33,40,36,43)>;
G:=Group( (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,36)(12,35)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39), (1,2,3,4,5,6,7,8,9,10,11,12)(13,18,23,16,21,14,19,24,17,22,15,20)(25,36,35,34,33,32,31,30,29,28,27,26)(37,44,39,46,41,48,43,38,45,40,47,42), (1,22,10,19,7,16,4,13)(2,23,11,20,8,17,5,14)(3,24,12,21,9,18,6,15)(25,44,28,47,31,38,34,41)(26,45,29,48,32,39,35,42)(27,46,30,37,33,40,36,43) );
G=PermutationGroup([(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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,36),(12,35),(13,38),(14,37),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,18,23,16,21,14,19,24,17,22,15,20),(25,36,35,34,33,32,31,30,29,28,27,26),(37,44,39,46,41,48,43,38,45,40,47,42)], [(1,22,10,19,7,16,4,13),(2,23,11,20,8,17,5,14),(3,24,12,21,9,18,6,15),(25,44,28,47,31,38,34,41),(26,45,29,48,32,39,35,42),(27,46,30,37,33,40,36,43)])
54 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | ··· | 12F | 12G | ··· | 12K | 12L | 12M | 24A | ··· | 24H |
order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
size | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 4 | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 3 | 3 | 3 | 3 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 6 | ··· | 6 |
54 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | - | + | - | - | + | + | - | + | - | |||||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | S3 | D6 | Dic3 | D6 | Dic3 | Dic3 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | S32 | S3×Dic3 | C2×S32 | S3×Dic3 | D4.Dic3 | D12.2Dic3 |
kernel | D12.2Dic3 | S3×C3⋊C8 | D6.Dic3 | C6×C3⋊C8 | C12.58D6 | C3×C4○D12 | C3×Dic6 | C3×D12 | C3×C3⋊D4 | C2×C3⋊C8 | C4○D12 | C3⋊C8 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C12 | C2×C6 | C32 | C3 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
# reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 4 |
Matrix representation of D12.2Dic3 ►in GL6(𝔽73)
1 | 72 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 46 | 0 | 0 | 0 |
0 | 0 | 46 | 27 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
1 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 27 | 19 | 0 | 0 |
0 | 0 | 27 | 46 | 0 | 0 |
0 | 0 | 0 | 0 | 72 | 0 |
0 | 0 | 0 | 0 | 0 | 72 |
72 | 0 | 0 | 0 | 0 | 0 |
0 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 46 | 0 | 0 | 0 |
0 | 0 | 0 | 46 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 72 | 1 |
27 | 0 | 0 | 0 | 0 | 0 |
0 | 27 | 0 | 0 | 0 | 0 |
0 | 0 | 22 | 0 | 0 | 0 |
0 | 0 | 0 | 22 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 46 |
0 | 0 | 0 | 0 | 46 | 0 |
G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,46,46,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,27,27,0,0,0,0,19,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;
D12.2Dic3 in GAP, Magma, Sage, TeX
D_{12}._2{\rm Dic}_3
% in TeX
G:=Group("D12.2Dic3");
// GroupNames label
G:=SmallGroup(288,462);
// by ID
G=gap.SmallGroup(288,462);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^6*c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations