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## G = Dic3.D8order 192 = 26·3

### 1st non-split extension by Dic3 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic3.D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — D4×Dic3 — Dic3.D8
 Lower central C3 — C6 — C2×C12 — Dic3.D8
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic3.D8
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=a3c-1 >

Subgroups: 312 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, C22×Dic3, C6×D4, D4⋊Q8, C6.Q16, Dic3⋊C8, C241C4, D4⋊Dic3, C3×D4⋊C4, C12⋊Q8, D4×Dic3, Dic3.D8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C2×D8, C8.C22, C2×Dic6, S3×D4, D42S3, D4⋊Q8, Dic3.D4, S3×D8, D4.D6, Dic3.D8

Smallest permutation representation of Dic3.D8
On 96 points
Generators in S96
(1 69 59 81 42 9)(2 70 60 82 43 10)(3 71 61 83 44 11)(4 72 62 84 45 12)(5 65 63 85 46 13)(6 66 64 86 47 14)(7 67 57 87 48 15)(8 68 58 88 41 16)(17 80 51 94 27 37)(18 73 52 95 28 38)(19 74 53 96 29 39)(20 75 54 89 30 40)(21 76 55 90 31 33)(22 77 56 91 32 34)(23 78 49 92 25 35)(24 79 50 93 26 36)
(1 18 81 95)(2 96 82 19)(3 20 83 89)(4 90 84 21)(5 22 85 91)(6 92 86 23)(7 24 87 93)(8 94 88 17)(9 73 59 28)(10 29 60 74)(11 75 61 30)(12 31 62 76)(13 77 63 32)(14 25 64 78)(15 79 57 26)(16 27 58 80)(33 72 55 45)(34 46 56 65)(35 66 49 47)(36 48 50 67)(37 68 51 41)(38 42 52 69)(39 70 53 43)(40 44 54 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)
(2 88)(3 7)(4 86)(6 84)(8 82)(10 58)(11 15)(12 64)(14 62)(16 60)(17 96)(19 94)(20 24)(21 92)(23 90)(25 76)(26 30)(27 74)(29 80)(31 78)(33 49)(35 55)(36 40)(37 53)(39 51)(41 70)(43 68)(44 48)(45 66)(47 72)(50 54)(57 61)(67 71)(75 79)(83 87)(89 93)

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

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

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

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 4 4 2 2 2 6 6 8 12 12 12 24 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 Q8 D6 D6 D6 D8 C4○D4 Dic6 C8.C22 D4⋊2S3 S3×D4 S3×D8 D4.D6 kernel Dic3.D8 C6.Q16 Dic3⋊C8 C24⋊1C4 D4⋊Dic3 C3×D4⋊C4 C12⋊Q8 D4×Dic3 D4⋊C4 C2×Dic3 C3×D4 C4⋊C4 C2×C8 C2×D4 Dic3 C12 D4 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 Dic3.D8 in GL4(𝔽73) generated by

 0 72 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 30 43 0 0 13 43 0 0 0 0 72 0 0 0 0 72
,
 7 14 0 0 59 66 0 0 0 0 16 57 0 0 16 16
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [0,1,0,0,72,1,0,0,0,0,1,0,0,0,0,1],[30,13,0,0,43,43,0,0,0,0,72,0,0,0,0,72],[7,59,0,0,14,66,0,0,0,0,16,16,0,0,57,16],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;

Dic3.D8 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_8
% in TeX

G:=Group("Dic3.D8");
// GroupNames label

G:=SmallGroup(192,318);
// by ID

G=gap.SmallGroup(192,318);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,254,219,58,851,438,102,6278]);
// 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=a^3*b,b*d=d*b,d*c*d=a^3*c^-1>;
// generators/relations

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