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## G = Dic3⋊5SD16order 192 = 26·3

### 2nd semidirect product of Dic3 and SD16 acting via SD16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic3⋊5SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C12⋊3D4 — Dic3⋊5SD16
 Lower central C3 — C6 — C2×C12 — Dic3⋊5SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for Dic35SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c3 >

Subgroups: 424 in 128 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16, C2×SD16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, Q82S3, C2×C24, C3×SD16, C2×D12, C2×C3⋊D4, C6×D4, C6×Q8, C4⋊SD16, Dic3⋊C8, C2.D24, D4⋊Dic3, C123D4, C2×Q82S3, Q8×Dic3, C6×SD16, Dic35SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8⋊C22, S3×D4, D42S3, C2×C3⋊D4, C4⋊SD16, S3×SD16, Q83D6, C23.14D6, Dic35SD16

Smallest permutation representation of Dic35SD16
On 96 points
Generators in S96
(1 73 30 14 49 83)(2 74 31 15 50 84)(3 75 32 16 51 85)(4 76 25 9 52 86)(5 77 26 10 53 87)(6 78 27 11 54 88)(7 79 28 12 55 81)(8 80 29 13 56 82)(17 89 63 35 43 67)(18 90 64 36 44 68)(19 91 57 37 45 69)(20 92 58 38 46 70)(21 93 59 39 47 71)(22 94 60 40 48 72)(23 95 61 33 41 65)(24 96 62 34 42 66)
(1 18 14 36)(2 37 15 19)(3 20 16 38)(4 39 9 21)(5 22 10 40)(6 33 11 23)(7 24 12 34)(8 35 13 17)(25 93 86 47)(26 48 87 94)(27 95 88 41)(28 42 81 96)(29 89 82 43)(30 44 83 90)(31 91 84 45)(32 46 85 92)(49 64 73 68)(50 69 74 57)(51 58 75 70)(52 71 76 59)(53 60 77 72)(54 65 78 61)(55 62 79 66)(56 67 80 63)
(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 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 33)(10 36)(11 39)(12 34)(13 37)(14 40)(15 35)(16 38)(25 61)(26 64)(27 59)(28 62)(29 57)(30 60)(31 63)(32 58)(41 52)(42 55)(43 50)(44 53)(45 56)(46 51)(47 54)(48 49)(65 86)(66 81)(67 84)(68 87)(69 82)(70 85)(71 88)(72 83)(73 94)(74 89)(75 92)(76 95)(77 90)(78 93)(79 96)(80 91)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 1 1 8 24 2 2 2 4 4 6 6 12 12 12 2 2 2 8 8 4 4 12 12 4 4 8 8 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 SD16 C4○D4 C3⋊D4 C8⋊C22 D4⋊2S3 S3×D4 S3×SD16 Q8⋊3D6 kernel Dic3⋊5SD16 Dic3⋊C8 C2.D24 D4⋊Dic3 C12⋊3D4 C2×Q8⋊2S3 Q8×Dic3 C6×SD16 C2×SD16 C2×Dic3 C3×Q8 C2×C8 C2×D4 C2×Q8 Dic3 C12 Q8 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 2 4 1 1 1 2 2

Matrix representation of Dic35SD16 in GL4(𝔽73) generated by

 1 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 60 43 0 0 30 13 0 0 0 0 72 0 0 0 0 72
,
 30 13 0 0 60 43 0 0 0 0 61 55 0 0 4 0
,
 43 60 0 0 13 30 0 0 0 0 1 0 0 0 48 72
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[60,30,0,0,43,13,0,0,0,0,72,0,0,0,0,72],[30,60,0,0,13,43,0,0,0,0,61,4,0,0,55,0],[43,13,0,0,60,30,0,0,0,0,1,48,0,0,0,72] >;

Dic35SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("Dic3:5SD16");
// GroupNames label

G:=SmallGroup(192,722);
// by ID

G=gap.SmallGroup(192,722);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,135,184,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations

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