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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 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C8, C2×C4, C2×C4, D4 [×3], C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×4], C2×C6 [×2], C2×C6 [×5], C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3×C6 [×3], C3⋊C8 [×4], D12 [×2], D12, C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×D4 [×3], C22×S3, C22×C6, D4⋊C4, C3×Dic3, C3×C12 [×2], S3×C6 [×4], C62, C2×C3⋊C8 [×3], C4⋊Dic3, C3×C4⋊C4, C2×D12, C6×D4, C324C8, C3×D12 [×2], C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C6.D8, D4⋊Dic3, C3×C4⋊Dic3, C2×C324C8, C6×D12, D123Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, D8, SD16, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], D4⋊C4, S32, D6⋊C4, D4⋊S3 [×2], 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 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 85)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 84)(46 83)(47 82)(48 81)
(1 23 9 19 5 15)(2 24 10 20 6 16)(3 13 11 21 7 17)(4 14 12 22 8 18)(25 39 29 43 33 47)(26 40 30 44 34 48)(27 41 31 45 35 37)(28 42 32 46 36 38)(49 64 53 68 57 72)(50 65 54 69 58 61)(51 66 55 70 59 62)(52 67 56 71 60 63)(73 88 81 96 77 92)(74 89 82 85 78 93)(75 90 83 86 79 94)(76 91 84 87 80 95)
(1 37 19 31)(2 44 20 26)(3 39 21 33)(4 46 22 28)(5 41 23 35)(6 48 24 30)(7 43 13 25)(8 38 14 32)(9 45 15 27)(10 40 16 34)(11 47 17 29)(12 42 18 36)(49 95 68 84)(50 90 69 79)(51 85 70 74)(52 92 71 81)(53 87 72 76)(54 94 61 83)(55 89 62 78)(56 96 63 73)(57 91 64 80)(58 86 65 75)(59 93 66 82)(60 88 67 77)

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

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

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

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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