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## G = Dic9.D6order 432 = 24·33

### 4th non-split extension by Dic9 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — Dic9.D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — C9⋊D12 — Dic9.D6
 Lower central C3×C9 — C3×C18 — Dic9.D6
 Upper central C1 — C2 — C4

Generators and relations for Dic9.D6
G = < a,b,c,d | a18=1, b2=c6=d2=a9, bab-1=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd-1=c5 >

Subgroups: 960 in 136 conjugacy classes, 41 normal (31 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×C9, Dic9, C36, C36, D18, C2×C18, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, Q83S3, S3×C9, C9⋊S3, C3×C18, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C12⋊S3, C3×Dic9, C9×Dic3, C3×C36, S3×C18, C2×C9⋊S3, D365C2, D6.6D6, C18.D6, C9⋊D12, C3×Dic18, S3×C36, C36⋊S3, Dic9.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, Q83S3, C22×D9, C2×S32, S3×D9, D365C2, D6.6D6, C2×S3×D9, Dic9.D6

Smallest permutation representation of Dic9.D6
On 72 points
Generators in S72
(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 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 51 10 42)(2 50 11 41)(3 49 12 40)(4 48 13 39)(5 47 14 38)(6 46 15 37)(7 45 16 54)(8 44 17 53)(9 43 18 52)(19 66 28 57)(20 65 29 56)(21 64 30 55)(22 63 31 72)(23 62 32 71)(24 61 33 70)(25 60 34 69)(26 59 35 68)(27 58 36 67)
(1 31 4 34 7 19 10 22 13 25 16 28)(2 32 5 35 8 20 11 23 14 26 17 29)(3 33 6 36 9 21 12 24 15 27 18 30)(37 67 52 64 49 61 46 58 43 55 40 70)(38 68 53 65 50 62 47 59 44 56 41 71)(39 69 54 66 51 63 48 60 45 57 42 72)
(1 39 10 48)(2 40 11 49)(3 41 12 50)(4 42 13 51)(5 43 14 52)(6 44 15 53)(7 45 16 54)(8 46 17 37)(9 47 18 38)(19 69 28 60)(20 70 29 61)(21 71 30 62)(22 72 31 63)(23 55 32 64)(24 56 33 65)(25 57 34 66)(26 58 35 67)(27 59 36 68)

G:=sub<Sym(72)| (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,51,10,42)(2,50,11,41)(3,49,12,40)(4,48,13,39)(5,47,14,38)(6,46,15,37)(7,45,16,54)(8,44,17,53)(9,43,18,52)(19,66,28,57)(20,65,29,56)(21,64,30,55)(22,63,31,72)(23,62,32,71)(24,61,33,70)(25,60,34,69)(26,59,35,68)(27,58,36,67), (1,31,4,34,7,19,10,22,13,25,16,28)(2,32,5,35,8,20,11,23,14,26,17,29)(3,33,6,36,9,21,12,24,15,27,18,30)(37,67,52,64,49,61,46,58,43,55,40,70)(38,68,53,65,50,62,47,59,44,56,41,71)(39,69,54,66,51,63,48,60,45,57,42,72), (1,39,10,48)(2,40,11,49)(3,41,12,50)(4,42,13,51)(5,43,14,52)(6,44,15,53)(7,45,16,54)(8,46,17,37)(9,47,18,38)(19,69,28,60)(20,70,29,61)(21,71,30,62)(22,72,31,63)(23,55,32,64)(24,56,33,65)(25,57,34,66)(26,58,35,67)(27,59,36,68)>;

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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,51,10,42)(2,50,11,41)(3,49,12,40)(4,48,13,39)(5,47,14,38)(6,46,15,37)(7,45,16,54)(8,44,17,53)(9,43,18,52)(19,66,28,57)(20,65,29,56)(21,64,30,55)(22,63,31,72)(23,62,32,71)(24,61,33,70)(25,60,34,69)(26,59,35,68)(27,58,36,67), (1,31,4,34,7,19,10,22,13,25,16,28)(2,32,5,35,8,20,11,23,14,26,17,29)(3,33,6,36,9,21,12,24,15,27,18,30)(37,67,52,64,49,61,46,58,43,55,40,70)(38,68,53,65,50,62,47,59,44,56,41,71)(39,69,54,66,51,63,48,60,45,57,42,72), (1,39,10,48)(2,40,11,49)(3,41,12,50)(4,42,13,51)(5,43,14,52)(6,44,15,53)(7,45,16,54)(8,46,17,37)(9,47,18,38)(19,69,28,60)(20,70,29,61)(21,71,30,62)(22,72,31,63)(23,55,32,64)(24,56,33,65)(25,57,34,66)(26,58,35,67)(27,59,36,68) );

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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,51,10,42),(2,50,11,41),(3,49,12,40),(4,48,13,39),(5,47,14,38),(6,46,15,37),(7,45,16,54),(8,44,17,53),(9,43,18,52),(19,66,28,57),(20,65,29,56),(21,64,30,55),(22,63,31,72),(23,62,32,71),(24,61,33,70),(25,60,34,69),(26,59,35,68),(27,58,36,67)], [(1,31,4,34,7,19,10,22,13,25,16,28),(2,32,5,35,8,20,11,23,14,26,17,29),(3,33,6,36,9,21,12,24,15,27,18,30),(37,67,52,64,49,61,46,58,43,55,40,70),(38,68,53,65,50,62,47,59,44,56,41,71),(39,69,54,66,51,63,48,60,45,57,42,72)], [(1,39,10,48),(2,40,11,49),(3,41,12,50),(4,42,13,51),(5,43,14,52),(6,44,15,53),(7,45,16,54),(8,46,17,37),(9,47,18,38),(19,69,28,60),(20,70,29,61),(21,71,30,62),(22,72,31,63),(23,55,32,64),(24,56,33,65),(25,57,34,66),(26,58,35,67),(27,59,36,68)]])

63 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I 18A 18B 18C 18D 18E 18F 18G ··· 18L 36A ··· 36F 36G ··· 36L 36M ··· 36R order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 6 54 54 2 2 4 2 3 3 18 18 2 2 4 6 6 2 2 2 4 4 4 2 2 4 4 4 6 6 36 36 2 2 2 4 4 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 C4○D4 D9 D18 D18 D18 C4○D12 D36⋊5C2 S32 Q8⋊3S3 C2×S32 S3×D9 D6.6D6 C2×S3×D9 Dic9.D6 kernel Dic9.D6 C18.D6 C9⋊D12 C3×Dic18 S3×C36 C36⋊S3 Dic18 S3×C12 Dic9 C36 C3×Dic3 C3×C12 S3×C6 C3×C9 C4×S3 Dic3 C12 D6 C32 C3 C12 C9 C6 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 2 1 1 1 1 2 3 3 3 3 4 12 1 1 1 3 2 3 6

Matrix representation of Dic9.D6 in GL6(𝔽37)

 0 36 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 26 31 0 0 0 0 6 20
,
 6 0 0 0 0 0 31 31 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 5 10 0 0 0 0 27 32 0 0 0 0 0 0 0 1 0 0 0 0 36 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 0 6 0 0 0 0 0 0 0 36 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(37))| [0,1,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,6,0,0,0,0,31,20],[6,31,0,0,0,0,0,31,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,27,0,0,0,0,10,32,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic9.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_9.D_6
% in TeX

G:=Group("Dic9.D6");
// GroupNames label

G:=SmallGroup(432,289);
// by ID

G=gap.SmallGroup(432,289);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,135,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^18=1,b^2=c^6=d^2=a^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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