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G = D123Dic3order 288 = 25·32

3rd semidirect product of D12 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: D123Dic3, C12.15D12, C62.17D4, (C3×D12)⋊1C4, (C3×C6).13D8, C12.13(C4×S3), C4⋊Dic310S3, (C6×D12).2C2, (C2×D12).6S3, (C2×C12).75D6, (C3×C12).32D4, (C3×C6).7SD16, C4.6(S3×Dic3), C6.34(D6⋊C4), C6.11(D4⋊S3), C33(C6.D8), C6.4(D4.S3), C12.7(C2×Dic3), C32(D4⋊Dic3), C12.28(C3⋊D4), C324(D4⋊C4), C2.5(D6⋊Dic3), (C6×C12).24C22, C6.4(Q82S3), C2.1(C322D8), C4.21(C3⋊D12), C6.4(C6.D4), C2.1(Dic6⋊S3), C22.7(D6⋊S3), (C2×C4).99S32, (C3×C4⋊Dic3)⋊1C2, (C3×C12).22(C2×C4), (C2×C324C8)⋊1C2, (C2×C6).48(C3⋊D4), (C3×C6).29(C22⋊C4), SmallGroup(288,210)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D123Dic3
C1C3C32C3×C6C62C6×C12C6×D12 — D123Dic3
C32C3×C6C3×C12 — D123Dic3
C1C22C2×C4

Generators and relations for D123Dic3
 G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 402 in 111 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3, C3×C6, C3⋊C8, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, C3×Dic3, C3×C12, S3×C6, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×D12, C6×D4, C324C8, C3×D12, C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C6.D8, D4⋊Dic3, C3×C4⋊Dic3, C2×C324C8, C6×D12, D123Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C4×S3, D12, C2×Dic3, C3⋊D4, D4⋊C4, S32, D6⋊C4, D4⋊S3, D4.S3, Q82S3, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C6.D8, D4⋊Dic3, C322D8, Dic6⋊S3, D6⋊Dic3, D123Dic3

Smallest permutation representation of D123Dic3
On 96 points
Generators in S96
(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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 72)(9 71)(10 70)(11 69)(12 68)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 60)(24 59)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 84)
(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 45 29 37 33 41)(26 46 30 38 34 42)(27 47 31 39 35 43)(28 48 32 40 36 44)(49 66 53 70 57 62)(50 67 54 71 58 63)(51 68 55 72 59 64)(52 69 56 61 60 65)(73 94 81 90 77 86)(74 95 82 91 78 87)(75 96 83 92 79 88)(76 85 84 93 80 89)
(1 43 13 31)(2 38 14 26)(3 45 15 33)(4 40 16 28)(5 47 17 35)(6 42 18 30)(7 37 19 25)(8 44 20 32)(9 39 21 27)(10 46 22 34)(11 41 23 29)(12 48 24 36)(49 88 70 83)(50 95 71 78)(51 90 72 73)(52 85 61 80)(53 92 62 75)(54 87 63 82)(55 94 64 77)(56 89 65 84)(57 96 66 79)(58 91 67 74)(59 86 68 81)(60 93 69 76)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A···6F6G6H6I6J6K6L6M8A8B8C8D12A···12H12I12J12K12L
order12222233344446···66666666888812···1212121212
size111112122242212122···244412121212181818184···412121212

42 irreducible representations

dim11111222222222222444444444
type++++++++-+++++-+-+-
imageC1C2C2C2C4S3S3D4D4Dic3D6D8SD16C4×S3D12C3⋊D4C3⋊D4S32D4⋊S3D4.S3Q82S3S3×Dic3C3⋊D12D6⋊S3C322D8Dic6⋊S3
kernelD123Dic3C3×C4⋊Dic3C2×C324C8C6×D12C3×D12C4⋊Dic3C2×D12C3×C12C62D12C2×C12C3×C6C3×C6C12C12C12C2×C6C2×C4C6C6C6C4C4C22C2C2
# reps11114111122222224121111122

Matrix representation of D123Dic3 in GL6(𝔽73)

2700000
71460000
001000
000100
0000172
000010
,
3840000
59350000
0072000
0007200
00005448
00002919
,
7200000
0720000
001100
0072000
000010
000001
,
40580000
24330000
0046000
00272700
00003013
00006043

G:=sub<GL(6,GF(73))| [27,71,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[38,59,0,0,0,0,4,35,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,54,29,0,0,0,0,48,19],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,24,0,0,0,0,58,33,0,0,0,0,0,0,46,27,0,0,0,0,0,27,0,0,0,0,0,0,30,60,0,0,0,0,13,43] >;

D123Dic3 in GAP, Magma, Sage, TeX

D_{12}\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("D12:3Dic3");
// GroupNames label

G:=SmallGroup(288,210);
// by ID

G=gap.SmallGroup(288,210);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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