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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, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×D4, C4○D4, C4○D4, C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, 2+ 1+4, C3×Dic3, C3×C12, C4.A4, C4.A4, C2×D12, C4○D12, S3×D4, Q83S3, C3×C4○D4, C3×SL2(𝔽3), C3×Dic6, Q8.A4, D4○D12, Dic3.A4, C3×C4.A4, Dic6.A4
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C2×A4, 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 18 7 24)(2 17 8 23)(3 16 9 22)(4 15 10 21)(5 14 11 20)(6 13 12 19)(25 62 31 68)(26 61 32 67)(27 72 33 66)(28 71 34 65)(29 70 35 64)(30 69 36 63)(37 58 43 52)(38 57 44 51)(39 56 45 50)(40 55 46 49)(41 54 47 60)(42 53 48 59)
(1 15 7 21)(2 16 8 22)(3 17 9 23)(4 18 10 24)(5 19 11 13)(6 20 12 14)(25 68 31 62)(26 69 32 63)(27 70 33 64)(28 71 34 65)(29 72 35 66)(30 61 36 67)(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 16 19 22)(14 17 20 23)(15 18 21 24)(25 71 31 65)(26 72 32 66)(27 61 33 67)(28 62 34 68)(29 63 35 69)(30 64 36 70)(37 56 43 50)(38 57 44 51)(39 58 45 52)(40 59 46 53)(41 60 47 54)(42 49 48 55)
(1 31 47)(2 32 48)(3 33 37)(4 34 38)(5 35 39)(6 36 40)(7 25 41)(8 26 42)(9 27 43)(10 28 44)(11 29 45)(12 30 46)(13 63 55)(14 64 56)(15 65 57)(16 66 58)(17 67 59)(18 68 60)(19 69 49)(20 70 50)(21 71 51)(22 72 52)(23 61 53)(24 62 54)

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

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

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

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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