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G = A4×Dic6order 288 = 25·32

Direct product of A4 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — A4×Dic6
 Chief series C1 — C3 — C2×C6 — C22×C6 — C6×A4 — Dic3×A4 — A4×Dic6
 Lower central C2×C6 — C22×C6 — A4×Dic6
 Upper central C1 — C2 — C4

Generators and relations for A4×Dic6
G = < a,b,c,d,e | a2=b2=c3=d12=1, e2=d6, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 378 in 98 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4, C4 [×5], C22, C22 [×2], C6, C6 [×4], C2×C4 [×6], Q8 [×6], C23, C32, Dic3 [×2], Dic3 [×2], C12, C12 [×5], A4, A4, C2×C6, C2×C6 [×2], C22×C4, C22×C4 [×2], C2×Q8 [×4], C3×C6, Dic6, Dic6 [×5], C2×Dic3 [×4], C2×C12 [×2], C3×Q8, C2×A4, C2×A4, C22×C6, C22×Q8, C3×Dic3 [×2], C3×C12, C3×A4, C4×A4, C4×A4 [×3], C2×Dic6 [×4], C22×Dic3 [×2], C22×C12, C3×Dic6, C6×A4, Q8×A4, C22×Dic6, Dic3×A4 [×2], C12×A4, A4×Dic6
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], Q8, A4, D6, C2×C6, C3×S3, Dic6, C3×Q8, C2×A4 [×3], S3×C6, C22×A4, C3×Dic6, S3×A4, Q8×A4, C2×S3×A4, A4×Dic6

Smallest permutation representation of A4×Dic6
On 72 points
Generators in S72
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
(1 19 68)(2 20 69)(3 21 70)(4 22 71)(5 23 72)(6 24 61)(7 13 62)(8 14 63)(9 15 64)(10 16 65)(11 17 66)(12 18 67)(25 38 52)(26 39 53)(27 40 54)(28 41 55)(29 42 56)(30 43 57)(31 44 58)(32 45 59)(33 46 60)(34 47 49)(35 48 50)(36 37 51)
(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 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)(49 61 55 67)(50 72 56 66)(51 71 57 65)(52 70 58 64)(53 69 59 63)(54 68 60 62)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 6 6 6 6 type + + + + - + - + + + + - + - image C1 C2 C2 C3 C6 C6 S3 Q8 D6 C3×S3 Dic6 C3×Q8 S3×C6 C3×Dic6 A4 C2×A4 C2×A4 S3×A4 Q8×A4 C2×S3×A4 A4×Dic6 kernel A4×Dic6 Dic3×A4 C12×A4 C22×Dic6 C22×Dic3 C22×C12 C4×A4 C3×A4 C2×A4 C22×C4 A4 C2×C6 C23 C22 Dic6 Dic3 C12 C4 C3 C2 C1 # reps 1 2 1 2 4 2 1 1 1 2 2 2 2 4 1 2 1 1 1 1 2

Matrix representation of A4×Dic6 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 12 12 12 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 12 12 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 10 10 10
,
 6 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 12 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,12,0,0,0,1,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,10,0,0,0,0,10,0,0,0,3,10],[6,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12] >;

A4×Dic6 in GAP, Magma, Sage, TeX

A_4\times {\rm Dic}_6
% in TeX

G:=Group("A4xDic6");
// GroupNames label

G:=SmallGroup(288,918);
// by ID

G=gap.SmallGroup(288,918);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,84,197,92,648,271,9414]);
// Polycyclic

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

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