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## G = Dic6.A4order 288 = 25·32

### The non-split extension by Dic6 of A4 acting through Inn(Dic6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — Dic6.A4
 Chief series C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — Dic3.A4 — Dic6.A4
 Lower central C3×Q8 — Dic6.A4
 Upper central C1 — C2 — C4

Generators and relations for Dic6.A4
G = < a,b,c,d,e | a12=e3=1, b2=c2=d2=a6, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a6c, ece-1=a6cd, ede-1=c >

Subgroups: 510 in 97 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C4, C4 [×3], C22 [×5], S3 [×2], C6, C6 [×3], C2×C4 [×3], D4 [×6], Q8, Q8, C23 [×2], C32, Dic3 [×2], C12, C12 [×5], D6 [×4], C2×C6, C2×D4 [×3], C4○D4, C4○D4 [×3], C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3 [×2], D12 [×3], C3⋊D4 [×2], C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3 [×2], 2+ 1+4, C3×Dic3 [×2], C3×C12, C4.A4, C4.A4 [×3], C2×D12, C4○D12, S3×D4 [×2], Q83S3 [×2], C3×C4○D4, C3×SL2(𝔽3), C3×Dic6, Q8.A4, D4○D12, Dic3.A4 [×2], C3×C4.A4, Dic6.A4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], A4, D6, C2×C6, C3×S3, C2×A4 [×3], S3×C6, C22×A4, S3×A4, Q8.A4, C2×S3×A4, Dic6.A4

Smallest permutation representation of Dic6.A4
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 35 7 29)(2 34 8 28)(3 33 9 27)(4 32 10 26)(5 31 11 25)(6 30 12 36)(13 69 19 63)(14 68 20 62)(15 67 21 61)(16 66 22 72)(17 65 23 71)(18 64 24 70)(37 54 43 60)(38 53 44 59)(39 52 45 58)(40 51 46 57)(41 50 47 56)(42 49 48 55)
(1 32 7 26)(2 33 8 27)(3 34 9 28)(4 35 10 29)(5 36 11 30)(6 25 12 31)(13 61 19 67)(14 62 20 68)(15 63 21 69)(16 64 22 70)(17 65 23 71)(18 66 24 72)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 52 55 58)(50 53 56 59)(51 54 57 60)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 61 22 67)(17 62 23 68)(18 63 24 69)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 58 43 52)(38 59 44 53)(39 60 45 54)(40 49 46 55)(41 50 47 56)(42 51 48 57)
(1 68 44)(2 69 45)(3 70 46)(4 71 47)(5 72 48)(6 61 37)(7 62 38)(8 63 39)(9 64 40)(10 65 41)(11 66 42)(12 67 43)(13 52 28)(14 53 29)(15 54 30)(16 55 31)(17 56 32)(18 57 33)(19 58 34)(20 59 35)(21 60 36)(22 49 25)(23 50 26)(24 51 27)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 12A 12B 12C ··· 12H 12I 12J 12K 12L 12M order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 12 12 12 ··· 12 12 12 12 12 12 size 1 1 6 18 18 2 4 4 8 8 2 6 6 6 2 4 4 8 8 12 2 2 8 ··· 8 12 24 24 24 24

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 4 4 4 4 6 6 type + + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 A4 C2×A4 C2×A4 Q8.A4 Q8.A4 Dic6.A4 Dic6.A4 S3×A4 C2×S3×A4 kernel Dic6.A4 Dic3.A4 C3×C4.A4 D4○D12 Q8⋊3S3 C3×C4○D4 C4.A4 SL2(𝔽3) C4○D4 Q8 Dic6 Dic3 C12 C3 C3 C1 C1 C4 C2 # reps 1 2 1 2 4 2 1 1 2 2 1 2 1 1 2 2 4 1 1

Matrix representation of Dic6.A4 in GL4(𝔽13) generated by

 7 10 0 0 3 10 0 0 0 0 7 10 0 0 3 10
,
 3 6 7 1 3 10 7 6 3 6 10 7 3 10 10 3
,
 1 0 11 0 0 1 0 11 1 0 12 0 0 1 0 12
,
 3 6 0 0 7 10 0 0 3 6 10 7 7 10 6 3
,
 4 0 8 11 0 4 2 10 8 12 0 0 1 9 0 0
G:=sub<GL(4,GF(13))| [7,3,0,0,10,10,0,0,0,0,7,3,0,0,10,10],[3,3,3,3,6,10,6,10,7,7,10,10,1,6,7,3],[1,0,1,0,0,1,0,1,11,0,12,0,0,11,0,12],[3,7,3,7,6,10,6,10,0,0,10,6,0,0,7,3],[4,0,8,1,0,4,12,9,8,2,0,0,11,10,0,0] >;

Dic6.A4 in GAP, Magma, Sage, TeX

{\rm Dic}_6.A_4
% in TeX

G:=Group("Dic6.A4");
// GroupNames label

G:=SmallGroup(288,924);
// by ID

G=gap.SmallGroup(288,924);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-2,1008,2045,1016,269,360,123,515,242,4037]);
// Polycyclic

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

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