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G = A4×Dic6order 288 = 25·32

Direct product of A4 and Dic6

direct product, metabelian, soluble, monomial

Aliases: A4×Dic6, C3⋊(Q8×A4), C4.1(S3×A4), (C3×A4)⋊4Q8, (C4×A4).3S3, C12.1(C2×A4), (C12×A4).3C2, (C2×A4).13D6, C6.1(C22×A4), (C22×Dic6)⋊C3, C23.16(S3×C6), (C22×C12).1C6, Dic3.1(C2×A4), (Dic3×A4).2C2, C222(C3×Dic6), (C22×Dic3).C6, (C6×A4).18C22, (C2×C6)⋊(C3×Q8), C2.3(C2×S3×A4), (C22×C4).3(C3×S3), (C22×C6).1(C2×C6), SmallGroup(288,918)

Series: Derived Chief Lower central Upper central

C1C22×C6 — A4×Dic6
C1C3C2×C6C22×C6C6×A4Dic3×A4 — A4×Dic6
C2×C6C22×C6 — A4×Dic6
C1C2C4

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 2A2B2C3A3B3C3D3E4A4B4C4D4E4F6A6B6C6D6E6F6G12A12B12C12D12E···12J12K12L12M12N
order12223333344444466666661212121212···1212121212
size11332448826661818244668822668···824242424

36 irreducible representations

dim111111222222223336666
type++++-+-++++-+-
imageC1C2C2C3C6C6S3Q8D6C3×S3Dic6C3×Q8S3×C6C3×Dic6A4C2×A4C2×A4S3×A4Q8×A4C2×S3×A4A4×Dic6
kernelA4×Dic6Dic3×A4C12×A4C22×Dic6C22×Dic3C22×C12C4×A4C3×A4C2×A4C22×C4A4C2×C6C23C22Dic6Dic3C12C4C3C2C1
# reps121242111222241211112

Matrix representation of A4×Dic6 in GL5(𝔽13)

10000
01000
00001
00121212
00100
,
10000
01000
00010
00100
00121212
,
10000
01000
00300
00003
00101010
,
60000
011000
00100
00010
00001
,
012000
10000
001200
000120
000012

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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