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## G = Dic3⋊C8⋊C2order 192 = 26·3

### 2nd semidirect product of Dic3⋊C8 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3⋊C8⋊C2
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C23.26D6 — Dic3⋊C8⋊C2
 Lower central C3 — C2×C6 — Dic3⋊C8⋊C2
 Upper central C1 — C2×C4 — C22×C8

Generators and relations for Dic3⋊C8⋊C2
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, dbd=bc4, cd=dc >

Subgroups: 216 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C2×C24, C22×C12, C42.6C22, Dic3⋊C8, C2×C4.Dic3, C23.26D6, C22×C24, Dic3⋊C8⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C8○D4, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C42.6C22, C8○D12, C2×Dic3⋊C4, Dic3⋊C8⋊C2

Smallest permutation representation of Dic3⋊C8⋊C2
On 96 points
Generators in S96
(1 19 64 87 50 78)(2 20 57 88 51 79)(3 21 58 81 52 80)(4 22 59 82 53 73)(5 23 60 83 54 74)(6 24 61 84 55 75)(7 17 62 85 56 76)(8 18 63 86 49 77)(9 37 45 93 28 67)(10 38 46 94 29 68)(11 39 47 95 30 69)(12 40 48 96 31 70)(13 33 41 89 32 71)(14 34 42 90 25 72)(15 35 43 91 26 65)(16 36 44 92 27 66)
(1 37 87 28)(2 29 88 38)(3 39 81 30)(4 31 82 40)(5 33 83 32)(6 25 84 34)(7 35 85 26)(8 27 86 36)(9 50 93 19)(10 20 94 51)(11 52 95 21)(12 22 96 53)(13 54 89 23)(14 24 90 55)(15 56 91 17)(16 18 92 49)(41 60 71 74)(42 75 72 61)(43 62 65 76)(44 77 66 63)(45 64 67 78)(46 79 68 57)(47 58 69 80)(48 73 70 59)
(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 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 81)(8 82)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(41 71)(42 72)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 73)(64 74)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 S3 D4 Q8 D6 D6 Dic6 C4×S3 C3⋊D4 C4×S3 C8○D4 C8○D12 kernel Dic3⋊C8⋊C2 Dic3⋊C8 C2×C4.Dic3 C23.26D6 C22×C24 C4⋊Dic3 C6.D4 C22×C8 C2×C12 C2×C12 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 4 4 1 2 2 2 1 4 2 4 2 8 16

Matrix representation of Dic3⋊C8⋊C2 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 65 0 0 0 0 9
,
 0 1 0 0 1 0 0 0 0 0 0 72 0 0 1 0
,
 63 0 0 0 0 63 0 0 0 0 63 0 0 0 0 10
,
 72 0 0 0 0 1 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,65,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,72,0],[63,0,0,0,0,63,0,0,0,0,63,0,0,0,0,10],[72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;

Dic3⋊C8⋊C2 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes C_8\rtimes C_2
% in TeX

G:=Group("Dic3:C8:C2");
// GroupNames label

G:=SmallGroup(192,661);
// by ID

G=gap.SmallGroup(192,661);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,422,58,136,6278]);
// 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=a^3*b,d*b*d=b*c^4,c*d=d*c>;
// generators/relations

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