Copied to
clipboard

## G = D12⋊2Dic3order 288 = 25·32

### 2nd semidirect product of D12 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12⋊2Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C3×C4○D12 — D12⋊2Dic3
 Lower central C32 — C3×C6 — C3×C12 — D12⋊2Dic3
 Upper central C1 — C4 — C2×C4

Generators and relations for D122Dic3
G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 306 in 98 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×5], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C32, Dic3 [×3], C12 [×4], C12 [×5], D6, C2×C6 [×2], C2×C6 [×2], C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8 [×4], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C3×Q8, C4≀C2, C3×Dic3 [×3], C3×C12 [×2], S3×C6, C62, C4.Dic3 [×3], C4×Dic3, C4×C12, C4○D12, C3×C4○D4, C324C8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, C424S3, Q83Dic3, Dic3×C12, C12.58D6, C3×C4○D12, D122Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], C4≀C2, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C424S3, Q83Dic3, D6⋊Dic3, D122Dic3

Smallest permutation representation of D122Dic3
On 48 points
Generators in S48
(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 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 47 45 43 41 39)(38 48 46 44 42 40)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 47 31 41)(26 48 32 42)(27 37 33 43)(28 38 34 44)(29 39 35 45)(30 40 36 46)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 12A ··· 12F 12G ··· 12K 12L ··· 12S 12T 12U order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 8 8 12 ··· 12 12 ··· 12 12 ··· 12 12 12 size 1 1 2 12 2 2 4 1 1 2 6 6 6 6 12 2 2 2 2 4 4 4 4 12 12 36 36 2 ··· 2 4 ··· 4 6 ··· 6 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - - + + + - + - image C1 C2 C2 C2 C4 C4 S3 S3 D4 D4 Dic3 Dic3 D6 C4×S3 D12 C3⋊D4 C3⋊D4 C4≀C2 C42⋊4S3 S32 S3×Dic3 C3⋊D12 D6⋊S3 Q8⋊3Dic3 D12⋊2Dic3 kernel D12⋊2Dic3 Dic3×C12 C12.58D6 C3×C4○D12 C3×Dic6 C3×D12 C4×Dic3 C4○D12 C3×C12 C62 Dic6 D12 C2×C12 C12 C12 C12 C2×C6 C32 C3 C2×C4 C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 2 4 4 8 1 1 1 1 2 4

Matrix representation of D122Dic3 in GL6(𝔽73)

 27 0 0 0 0 0 0 46 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 46 0 0 0 0 27 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 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 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 46 46

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

D122Dic3 in GAP, Magma, Sage, TeX

D_{12}\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("D12:2Dic3");
// GroupNames label

G:=SmallGroup(288,217);
// by ID

G=gap.SmallGroup(288,217);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽