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## G = Dic6.20D6order 288 = 25·32

### 7th non-split extension by Dic6 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6.20D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.6D6 — Dic6.20D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6.20D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic6.20D6
G = < a,b,c,d | a12=c6=1, b2=d2=a6, bab-1=a-1, cac-1=dad-1=a7, cbc-1=a9b, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 546 in 135 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×4], C6 [×2], C6 [×5], C8 [×2], C2×C4 [×3], D4, D4 [×3], Q8 [×2], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×4], D6, D6 [×3], C2×C6 [×4], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6 [×2], C4×S3, C4×S3 [×2], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4○D8, C3×Dic3, C3×Dic3 [×2], C3×C12, S3×C6, C2×C3⋊S3, C62, S3×C8, C24⋊C2, C2×C3⋊C8, D4⋊S3 [×3], D4.S3, Q82S3, C3⋊Q16 [×2], C3×SD16, C4○D12, D42S3, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, C6.D6, C3⋊D12, C3×Dic6 [×2], S3×C12, C6×Dic3, C3×C3⋊D4, C12⋊S3, D4×C32, Q8.7D6, Q8.13D6, S3×C3⋊C8, C325SD16, C322Q16, C3×D4.S3, C327D8, D6.6D6, C3×D42S3, Dic6.20D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C4○D8, S32, S3×D4, C2×C3⋊D4, C2×S32, Q8.7D6, Q8.13D6, S3×C3⋊D4, Dic6.20D6

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

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

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

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

36 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 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 24 size 1 1 4 6 36 2 2 4 2 3 3 12 12 2 2 4 4 4 8 8 8 12 6 6 18 18 4 4 6 6 8 12 12 24 12 12

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 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 C4○D8 S32 S3×D4 C2×S32 Q8.7D6 Q8.13D6 S3×C3⋊D4 Dic6.20D6 kernel Dic6.20D6 S3×C3⋊C8 C32⋊5SD16 C32⋊2Q16 C3×D4.S3 C32⋊7D8 D6.6D6 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 4 1 1 1 2 2 2 1

Matrix representation of Dic6.20D6 in GL6(𝔽73)

 1 48 0 0 0 0 3 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 27 55 0 0 0 0 0 46 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 1 1
,
 0 38 0 0 0 0 25 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 12 69 0 0 0 0 18 61 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,3,0,0,0,0,48,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[27,0,0,0,0,0,55,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1],[0,25,0,0,0,0,38,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[12,18,0,0,0,0,69,61,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic6.20D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{20}D_6
% in TeX

G:=Group("Dic6.20D6");
// GroupNames label

G:=SmallGroup(288,583);
// by ID

G=gap.SmallGroup(288,583);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,422,135,346,185,80,1356,9414]);
// 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=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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