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## G = D6⋊7Dic6order 288 = 25·32

### 3rd semidirect product of D6 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D6⋊7Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — D6⋊7Dic6
 Lower central C32 — C62 — D6⋊7Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for D67Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 578 in 167 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×11], C12 [×4], C12 [×5], 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 [×8], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×3], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×2], C6.D4 [×2], C3×C4⋊C4, C2×Dic6 [×3], S3×C2×C4, C22×C12, S3×C12 [×2], C6×Dic3, C6×Dic3 [×2], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, C4.D12, C12.48D4, D6⋊Dic3 [×2], Dic3⋊Dic3 [×2], C3×C4⋊Dic3, S3×C2×C12, C2×C324Q8, D67Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×2], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6, C2×D12, C4○D12, D42S3, S3×Q8, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C4.D12, C12.48D4, S3×Dic6, D125S3, C2×C3⋊D12, D67Dic6

Smallest permutation representation of D67Dic6
On 96 points
Generators in S96
(1 71 9 67 5 63)(2 72 10 68 6 64)(3 61 11 69 7 65)(4 62 12 70 8 66)(13 58 17 50 21 54)(14 59 18 51 22 55)(15 60 19 52 23 56)(16 49 20 53 24 57)(25 44 29 48 33 40)(26 45 30 37 34 41)(27 46 31 38 35 42)(28 47 32 39 36 43)(73 85 81 93 77 89)(74 86 82 94 78 90)(75 87 83 95 79 91)(76 88 84 96 80 92)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 80)(26 81)(27 82)(28 83)(29 84)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 85)(46 86)(47 87)(48 88)(49 72)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(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 82 7 76)(2 81 8 75)(3 80 9 74)(4 79 10 73)(5 78 11 84)(6 77 12 83)(13 48 19 42)(14 47 20 41)(15 46 21 40)(16 45 22 39)(17 44 23 38)(18 43 24 37)(25 56 31 50)(26 55 32 49)(27 54 33 60)(28 53 34 59)(29 52 35 58)(30 51 36 57)(61 96 67 90)(62 95 68 89)(63 94 69 88)(64 93 70 87)(65 92 71 86)(66 91 72 85)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 6 6 2 2 4 2 2 6 6 12 12 36 36 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + - + + + + - + - - + + - - image C1 C2 C2 C2 C2 C2 S3 S3 D4 Q8 D6 D6 D6 C4○D4 D12 C3⋊D4 Dic6 C4○D12 S32 D4⋊2S3 S3×Q8 C3⋊D12 C2×S32 S3×Dic6 D12⋊5S3 kernel D6⋊7Dic6 D6⋊Dic3 Dic3⋊Dic3 C3×C4⋊Dic3 S3×C2×C12 C2×C32⋊4Q8 C4⋊Dic3 S3×C2×C4 C3×C12 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 C12 D6 C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 1 2 2 3 2 1 2 4 4 4 4 1 1 1 2 1 2 2

Matrix representation of D67Dic6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 2 9 0 0 0 0 4 11 0 0 0 0 0 0 2 4 0 0 0 0 9 11 0 0 0 0 0 0 12 0 0 0 0 0 12 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 10 3 0 0 0 0 10 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 5 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

D67Dic6 in GAP, Magma, Sage, TeX

D_6\rtimes_7{\rm Dic}_6
% in TeX

G:=Group("D6:7Dic6");
// GroupNames label

G:=SmallGroup(288,505);
// by ID

G=gap.SmallGroup(288,505);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,219,100,1356,9414]);
// Polycyclic

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

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