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## G = Dic3.D12order 288 = 25·32

### 4th non-split extension by Dic3 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — Dic3.D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — Dic3.D12
 Lower central C32 — C62 — Dic3.D12
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic3.D12
G = < a,b,c,d | a6=c12=1, b2=d2=a3, bab-1=cac-1=dad-1=a-1, bc=cb, dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 762 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×6], C22, C22 [×6], S3 [×5], C6 [×6], C6 [×4], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×2], Dic3 [×5], C12 [×7], D6 [×13], C2×C6 [×2], C2×C6 [×4], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6 [×3], Dic6 [×4], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×S3 [×3], C22×C6, C4.4D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×3], C2×C3⋊S3 [×3], C62, C4×Dic3, D6⋊C4, D6⋊C4 [×6], C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6 [×2], C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C322Q8 [×2], C6×Dic3 [×3], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C427S3, C23.11D6, D6⋊Dic3, C6.D12, Dic3×C12, C3×D6⋊C4, C6.11D12, C2×C3⋊D12, C2×C322Q8, Dic3.D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C22×S3 [×2], C4.4D4, S32, C2×D12, C4○D12 [×3], S3×D4, D42S3, C2×S32, C427S3, C23.11D6, D6.D6, S3×D12, D6.3D6, Dic3.D12

Smallest permutation representation of Dic3.D12
On 48 points
Generators in S48
(1 45 9 41 5 37)(2 38 6 42 10 46)(3 47 11 43 7 39)(4 40 8 44 12 48)(13 29 17 33 21 25)(14 26 22 34 18 30)(15 31 19 35 23 27)(16 28 24 36 20 32)
(1 31 41 23)(2 32 42 24)(3 33 43 13)(4 34 44 14)(5 35 45 15)(6 36 46 16)(7 25 47 17)(8 26 48 18)(9 27 37 19)(10 28 38 20)(11 29 39 21)(12 30 40 22)
(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 30 41 22)(2 21 42 29)(3 28 43 20)(4 19 44 27)(5 26 45 18)(6 17 46 25)(7 36 47 16)(8 15 48 35)(9 34 37 14)(10 13 38 33)(11 32 39 24)(12 23 40 31)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 D12 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 D6.D6 S3×D12 D6.3D6 kernel Dic3.D12 D6⋊Dic3 C6.D12 Dic3×C12 C3×D6⋊C4 C6.11D12 C2×C3⋊D12 C2×C32⋊2Q8 C4×Dic3 D6⋊C4 C3×Dic3 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 3 2 1 4 4 12 1 1 1 1 2 2 2

Matrix representation of Dic3.D12 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

Dic3.D12 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,500);
// by ID

G=gap.SmallGroup(288,500);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,422,58,1356,9414]);
// Polycyclic

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

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