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## G = Dic3×M4(2)  order 192 = 26·3

### Direct product of Dic3 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Dic3×M4(2)
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — C2×C4×Dic3 — Dic3×M4(2)
 Lower central C3 — C6 — Dic3×M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for Dic3×M4(2)
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 248 in 142 conjugacy classes, 91 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×Dic3, C4×Dic3, C2×C24, C3×M4(2), C22×Dic3, C22×C12, C4×M4(2), C8×Dic3, C24⋊C4, C2×C4.Dic3, C2×C4×Dic3, C6×M4(2), Dic3×M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, M4(2), C22×C4, C4×S3, C2×Dic3, C22×S3, C2×C42, C2×M4(2), C4×Dic3, S3×C2×C4, C22×Dic3, C4×M4(2), S3×M4(2), C2×C4×Dic3, Dic3×M4(2)

Smallest permutation representation of Dic3×M4(2)
On 96 points
Generators in S96
(1 20 27 79 54 71)(2 21 28 80 55 72)(3 22 29 73 56 65)(4 23 30 74 49 66)(5 24 31 75 50 67)(6 17 32 76 51 68)(7 18 25 77 52 69)(8 19 26 78 53 70)(9 42 64 81 90 33)(10 43 57 82 91 34)(11 44 58 83 92 35)(12 45 59 84 93 36)(13 46 60 85 94 37)(14 47 61 86 95 38)(15 48 62 87 96 39)(16 41 63 88 89 40)
(1 58 79 35)(2 59 80 36)(3 60 73 37)(4 61 74 38)(5 62 75 39)(6 63 76 40)(7 64 77 33)(8 57 78 34)(9 69 81 25)(10 70 82 26)(11 71 83 27)(12 72 84 28)(13 65 85 29)(14 66 86 30)(15 67 87 31)(16 68 88 32)(17 41 51 89)(18 42 52 90)(19 43 53 91)(20 44 54 92)(21 45 55 93)(22 46 56 94)(23 47 49 95)(24 48 50 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 75)(2 80)(3 77)(4 74)(5 79)(6 76)(7 73)(8 78)(9 85)(10 82)(11 87)(12 84)(13 81)(14 86)(15 83)(16 88)(17 51)(18 56)(19 53)(20 50)(21 55)(22 52)(23 49)(24 54)(25 65)(26 70)(27 67)(28 72)(29 69)(30 66)(31 71)(32 68)(33 60)(34 57)(35 62)(36 59)(37 64)(38 61)(39 58)(40 63)(41 89)(42 94)(43 91)(44 96)(45 93)(46 90)(47 95)(48 92)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 6A 6B 6C 6D 6E 8A ··· 8H 8I ··· 8P 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 6 6 6 6 8 ··· 8 8 ··· 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 1 1 1 1 2 2 3 ··· 3 6 6 6 6 2 2 2 4 4 2 ··· 2 6 ··· 6 2 2 2 2 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D6 Dic3 D6 M4(2) C4×S3 C4×S3 S3×M4(2) kernel Dic3×M4(2) C8×Dic3 C24⋊C4 C2×C4.Dic3 C2×C4×Dic3 C6×M4(2) C4.Dic3 C4×Dic3 C3×M4(2) C22×Dic3 C2×M4(2) C2×C8 M4(2) C22×C4 Dic3 C2×C4 C23 C2 # reps 1 2 2 1 1 1 8 4 8 4 1 2 4 1 8 6 2 4

Matrix representation of Dic3×M4(2) in GL4(𝔽73) generated by

 0 72 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 14 19 0 0 5 59 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 27 14 0 0 19 46
,
 72 0 0 0 0 72 0 0 0 0 72 72 0 0 0 1
G:=sub<GL(4,GF(73))| [0,1,0,0,72,1,0,0,0,0,1,0,0,0,0,1],[14,5,0,0,19,59,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,27,19,0,0,14,46],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,72,1] >;

Dic3×M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_3\times M_4(2)
% in TeX

G:=Group("Dic3xM4(2)");
// GroupNames label

G:=SmallGroup(192,676);
// by ID

G=gap.SmallGroup(192,676);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,387,100,102,6278]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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