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G = D64Dic6order 288 = 25·32

4th semidirect product of D6 and Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial

Aliases: D64Dic6, C62.69C23, (S3×C6)⋊4Q8, D6⋊C4.9S3, C4⋊Dic38S3, C6.38(S3×D4), C6.30(S3×Q8), (C2×C12).26D6, C34(D6⋊Q8), D6⋊Dic3.7C2, C3⋊Dic3.17D4, C6.18(C2×Dic6), C2.18(S3×Dic6), C6.56(C4○D12), (C2×Dic3).29D6, C62.C228C2, (C22×S3).42D6, C2.14(D6⋊D6), C3213(C22⋊Q8), C6.43(D42S3), (C6×C12).186C22, C6.Dic611C2, C33(Dic3.D4), C2.19(D6.3D6), (C6×Dic3).14C22, (C2×C4).31S32, (C3×D6⋊C4).8C2, (C3×C6).54(C2×D4), (C3×C6).33(C2×Q8), (C2×S3×Dic3).9C2, (C3×C4⋊Dic3)⋊16C2, C22.112(C2×S32), (C2×C322Q8)⋊4C2, (S3×C2×C6).25C22, (C3×C6).68(C4○D4), (C2×C6).88(C22×S3), (C2×C3⋊Dic3).50C22, SmallGroup(288,547)

Series: Derived Chief Lower central Upper central

C1C62 — D64Dic6
C1C3C32C3×C6C62S3×C2×C6C2×S3×Dic3 — D64Dic6
C32C62 — D64Dic6
C1C22C2×C4

Generators and relations for D64Dic6
 G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 586 in 161 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×12], C12 [×6], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×S3 [×2], C3×C6 [×3], Dic6 [×4], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×3], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4 [×5], C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C322Q8 [×2], C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, Dic3.D4, D6⋊Q8, D6⋊Dic3, C62.C22, C3×C4⋊Dic3, C3×D6⋊C4, C6.Dic6, C2×S3×Dic3, C2×C322Q8, D64Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6, C4○D12, S3×D4 [×2], D42S3, S3×Q8, C2×S32, Dic3.D4, D6⋊Q8, S3×Dic6, D6⋊D6, D6.3D6, D64Dic6

Smallest permutation representation of D64Dic6
On 96 points
Generators in S96
(1 25 9 33 5 29)(2 30 6 34 10 26)(3 27 11 35 7 31)(4 32 8 36 12 28)(13 88 21 96 17 92)(14 93 18 85 22 89)(15 90 23 86 19 94)(16 95 20 87 24 91)(37 78 41 82 45 74)(38 75 46 83 42 79)(39 80 43 84 47 76)(40 77 48 73 44 81)(49 64 57 72 53 68)(50 69 54 61 58 65)(51 66 59 62 55 70)(52 71 56 63 60 67)
(1 41)(2 79)(3 43)(4 81)(5 45)(6 83)(7 47)(8 73)(9 37)(10 75)(11 39)(12 77)(13 69)(14 51)(15 71)(16 53)(17 61)(18 55)(19 63)(20 57)(21 65)(22 59)(23 67)(24 49)(25 78)(26 38)(27 80)(28 40)(29 82)(30 42)(31 84)(32 44)(33 74)(34 46)(35 76)(36 48)(50 88)(52 90)(54 92)(56 94)(58 96)(60 86)(62 85)(64 87)(66 89)(68 91)(70 93)(72 95)
(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 92 7 86)(2 91 8 85)(3 90 9 96)(4 89 10 95)(5 88 11 94)(6 87 12 93)(13 35 19 29)(14 34 20 28)(15 33 21 27)(16 32 22 26)(17 31 23 25)(18 30 24 36)(37 50 43 56)(38 49 44 55)(39 60 45 54)(40 59 46 53)(41 58 47 52)(42 57 48 51)(61 76 67 82)(62 75 68 81)(63 74 69 80)(64 73 70 79)(65 84 71 78)(66 83 72 77)

G:=sub<Sym(96)| (1,25,9,33,5,29)(2,30,6,34,10,26)(3,27,11,35,7,31)(4,32,8,36,12,28)(13,88,21,96,17,92)(14,93,18,85,22,89)(15,90,23,86,19,94)(16,95,20,87,24,91)(37,78,41,82,45,74)(38,75,46,83,42,79)(39,80,43,84,47,76)(40,77,48,73,44,81)(49,64,57,72,53,68)(50,69,54,61,58,65)(51,66,59,62,55,70)(52,71,56,63,60,67), (1,41)(2,79)(3,43)(4,81)(5,45)(6,83)(7,47)(8,73)(9,37)(10,75)(11,39)(12,77)(13,69)(14,51)(15,71)(16,53)(17,61)(18,55)(19,63)(20,57)(21,65)(22,59)(23,67)(24,49)(25,78)(26,38)(27,80)(28,40)(29,82)(30,42)(31,84)(32,44)(33,74)(34,46)(35,76)(36,48)(50,88)(52,90)(54,92)(56,94)(58,96)(60,86)(62,85)(64,87)(66,89)(68,91)(70,93)(72,95), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,92,7,86)(2,91,8,85)(3,90,9,96)(4,89,10,95)(5,88,11,94)(6,87,12,93)(13,35,19,29)(14,34,20,28)(15,33,21,27)(16,32,22,26)(17,31,23,25)(18,30,24,36)(37,50,43,56)(38,49,44,55)(39,60,45,54)(40,59,46,53)(41,58,47,52)(42,57,48,51)(61,76,67,82)(62,75,68,81)(63,74,69,80)(64,73,70,79)(65,84,71,78)(66,83,72,77)>;

G:=Group( (1,25,9,33,5,29)(2,30,6,34,10,26)(3,27,11,35,7,31)(4,32,8,36,12,28)(13,88,21,96,17,92)(14,93,18,85,22,89)(15,90,23,86,19,94)(16,95,20,87,24,91)(37,78,41,82,45,74)(38,75,46,83,42,79)(39,80,43,84,47,76)(40,77,48,73,44,81)(49,64,57,72,53,68)(50,69,54,61,58,65)(51,66,59,62,55,70)(52,71,56,63,60,67), (1,41)(2,79)(3,43)(4,81)(5,45)(6,83)(7,47)(8,73)(9,37)(10,75)(11,39)(12,77)(13,69)(14,51)(15,71)(16,53)(17,61)(18,55)(19,63)(20,57)(21,65)(22,59)(23,67)(24,49)(25,78)(26,38)(27,80)(28,40)(29,82)(30,42)(31,84)(32,44)(33,74)(34,46)(35,76)(36,48)(50,88)(52,90)(54,92)(56,94)(58,96)(60,86)(62,85)(64,87)(66,89)(68,91)(70,93)(72,95), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,92,7,86)(2,91,8,85)(3,90,9,96)(4,89,10,95)(5,88,11,94)(6,87,12,93)(13,35,19,29)(14,34,20,28)(15,33,21,27)(16,32,22,26)(17,31,23,25)(18,30,24,36)(37,50,43,56)(38,49,44,55)(39,60,45,54)(40,59,46,53)(41,58,47,52)(42,57,48,51)(61,76,67,82)(62,75,68,81)(63,74,69,80)(64,73,70,79)(65,84,71,78)(66,83,72,77) );

G=PermutationGroup([(1,25,9,33,5,29),(2,30,6,34,10,26),(3,27,11,35,7,31),(4,32,8,36,12,28),(13,88,21,96,17,92),(14,93,18,85,22,89),(15,90,23,86,19,94),(16,95,20,87,24,91),(37,78,41,82,45,74),(38,75,46,83,42,79),(39,80,43,84,47,76),(40,77,48,73,44,81),(49,64,57,72,53,68),(50,69,54,61,58,65),(51,66,59,62,55,70),(52,71,56,63,60,67)], [(1,41),(2,79),(3,43),(4,81),(5,45),(6,83),(7,47),(8,73),(9,37),(10,75),(11,39),(12,77),(13,69),(14,51),(15,71),(16,53),(17,61),(18,55),(19,63),(20,57),(21,65),(22,59),(23,67),(24,49),(25,78),(26,38),(27,80),(28,40),(29,82),(30,42),(31,84),(32,44),(33,74),(34,46),(35,76),(36,48),(50,88),(52,90),(54,92),(56,94),(58,96),(60,86),(62,85),(64,87),(66,89),(68,91),(70,93),(72,95)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,92,7,86),(2,91,8,85),(3,90,9,96),(4,89,10,95),(5,88,11,94),(6,87,12,93),(13,35,19,29),(14,34,20,28),(15,33,21,27),(16,32,22,26),(17,31,23,25),(18,30,24,36),(37,50,43,56),(38,49,44,55),(39,60,45,54),(40,59,46,53),(41,58,47,52),(42,57,48,51),(61,76,67,82),(62,75,68,81),(63,74,69,80),(64,73,70,79),(65,84,71,78),(66,83,72,77)])

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K12A···12H12I···12N
order122222333444444446···66666612···1212···12
size11116622446612121818362···244412124···412···12

42 irreducible representations

dim11111111222222222244444444
type+++++++++++-+++-++--+-
imageC1C2C2C2C2C2C2C2S3S3D4Q8D6D6D6C4○D4Dic6C4○D12S32S3×D4D42S3S3×Q8C2×S32S3×Dic6D6⋊D6D6.3D6
kernelD64Dic6D6⋊Dic3C62.C22C3×C4⋊Dic3C3×D6⋊C4C6.Dic6C2×S3×Dic3C2×C322Q8C4⋊Dic3D6⋊C4C3⋊Dic3S3×C6C2×Dic3C2×C12C22×S3C3×C6D6C6C2×C4C6C6C6C22C2C2C2
# reps11111111112232124412111222

Matrix representation of D64Dic6 in GL6(𝔽13)

1200000
0120000
001000
000100
0000012
000011
,
100000
12120000
001000
000100
0000310
0000710
,
830000
550000
000100
0012100
000008
000080
,
100000
010000
0001200
0012000
0000911
000024

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,7,0,0,0,0,10,10],[8,5,0,0,0,0,3,5,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,9,2,0,0,0,0,11,4] >;

D64Dic6 in GAP, Magma, Sage, TeX

D_6\rtimes_4{\rm Dic}_6
% in TeX

G:=Group("D6:4Dic6");
// GroupNames label

G:=SmallGroup(288,547);
// by ID

G=gap.SmallGroup(288,547);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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