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## G = Dic6.19D6order 288 = 25·32

### 6th non-split extension by Dic6 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6.19D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×Dic6 — Dic6.19D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6.19D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic6.19D6
G = < a,b,c,d | a12=c6=1, b2=d2=a6, bab-1=a-1, cac-1=dad-1=a7, cbc-1=dbd-1=a9b, dcd-1=a6c-1 >

Subgroups: 458 in 130 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C8⋊S3, Dic12, C4.Dic3, D4.S3, D4.S3, C3⋊Q16, C3×SD16, C2×Dic6, D42S3, S3×Q8, C3×C4○D4, C3×C3⋊C8, C324C8, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C324Q8, D4×C32, D4.D6, Q8.14D6, D6.Dic3, C322Q16, C323Q16, C3×D4.S3, C329SD16, S3×Dic6, C3×D42S3, Dic6.19D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, S32, S3×D4, C2×C3⋊D4, C2×S32, D4.D6, Q8.14D6, S3×C3⋊D4, Dic6.19D6

Smallest permutation representation of Dic6.19D6
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 13 7 19)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)(25 48 31 42)(26 47 32 41)(27 46 33 40)(28 45 34 39)(29 44 35 38)(30 43 36 37)
(1 5 9)(2 12 10 8 6 4)(3 7 11)(13 24 17 16 21 20)(14 19 18 23 22 15)(25 27 29 31 33 35)(26 34 30)(28 36 32)(37 44 45 40 41 48)(38 39 46 47 42 43)
(1 29 7 35)(2 36 8 30)(3 31 9 25)(4 26 10 32)(5 33 11 27)(6 28 12 34)(13 47 19 41)(14 42 20 48)(15 37 21 43)(16 44 22 38)(17 39 23 45)(18 46 24 40)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B order 1 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 12 12 12 12 12 12 12 12 24 24 size 1 1 4 6 2 2 4 2 6 12 12 36 2 2 4 4 4 8 8 8 12 12 36 4 4 6 6 8 12 12 24 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + - + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8.C22 S32 S3×D4 C2×S32 D4.D6 Q8.14D6 S3×C3⋊D4 Dic6.19D6 kernel Dic6.19D6 D6.Dic3 C32⋊2Q16 C32⋊3Q16 C3×D4.S3 C32⋊9SD16 S3×Dic6 C3×D4⋊2S3 D4.S3 D4⋊2S3 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 C3×D4 Dic3 D6 C32 D4 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 1 1 1 1 2 2 2 1

Matrix representation of Dic6.19D6 in GL8(𝔽73)

 64 0 0 0 0 0 0 0 44 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 29 64 0 0 0 0 0 0 0 0 1 66 0 0 0 0 0 0 42 72 0 0 0 0 0 0 0 0 72 7 0 0 0 0 0 0 31 1
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0
,
 64 0 0 0 0 0 0 0 44 8 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 44 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 42 72 0 0 0 0 0 0 0 0 72 7 0 0 0 0 0 0 0 1
,
 0 0 36 67 0 0 0 0 0 0 9 37 0 0 0 0 36 67 0 0 0 0 0 0 9 37 0 0 0 0 0 0 0 0 0 0 45 50 6 52 0 0 0 0 0 28 53 0 0 0 0 0 0 21 45 0 0 0 0 0 20 6 65 28

G:=sub<GL(8,GF(73))| [64,44,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,29,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,42,0,0,0,0,0,0,66,72,0,0,0,0,0,0,0,0,72,31,0,0,0,0,0,0,7,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[64,44,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,44,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,42,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,7,1],[0,0,36,9,0,0,0,0,0,0,67,37,0,0,0,0,36,9,0,0,0,0,0,0,67,37,0,0,0,0,0,0,0,0,0,0,45,0,0,20,0,0,0,0,50,28,21,6,0,0,0,0,6,53,45,65,0,0,0,0,52,0,0,28] >;

Dic6.19D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{19}D_6
% in TeX

G:=Group("Dic6.19D6");
// GroupNames label

G:=SmallGroup(288,577);
// by ID

G=gap.SmallGroup(288,577);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=1,b^2=d^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=d*b*d^-1=a^9*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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