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## G = Dic3.M4(2)  order 192 = 26·3

### 2nd non-split extension by Dic3 of M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3.M4(2)
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C2×C4×Dic3 — Dic3.M4(2)
 Lower central C3 — C2×C6 — Dic3.M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

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, cbc-1=a3b, bd=db, dcd=a3c5 >

Subgroups: 224 in 110 conjugacy classes, 53 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C2×C8, C2×C8, C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×Dic3, C2×C24, C22×Dic3, C22×C12, C42.6C4, Dic3⋊C8, C24⋊C4, C12.55D4, C3×C22⋊C8, C2×C4×Dic3, Dic3.M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C2×M4(2), C8⋊S3, S3×C2×C4, D42S3, C42.6C4, C23.16D6, C2×C8⋊S3, S3×M4(2), Dic3.M4(2)

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 S3 D6 D6 M4(2) C4○D4 M4(2) C4×S3 C4×S3 C8⋊S3 D4⋊2S3 S3×M4(2) kernel Dic3.M4(2) Dic3⋊C8 C24⋊C4 C12.55D4 C3×C22⋊C8 C2×C4×Dic3 C4×Dic3 C22×Dic3 C22⋊C8 C2×C8 C22×C4 Dic3 C12 C2×C6 C2×C4 C23 C22 C4 C2 # reps 1 2 2 1 1 1 4 4 1 2 1 4 4 4 2 2 8 2 2

Matrix representation of Dic3.M4(2) in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 46 0 0 0 0 0 25 27 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 23 55 0 0 0 0 5 50
,
 46 3 0 0 0 0 25 27 0 0 0 0 0 0 49 3 0 0 0 0 18 24 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 18 72 0 0 0 0 0 0 1 0 0 0 0 0 16 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,25,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,23,5,0,0,0,0,55,50],[46,25,0,0,0,0,3,27,0,0,0,0,0,0,49,18,0,0,0,0,3,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,18,0,0,0,0,0,72,0,0,0,0,0,0,1,16,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3.M4(2) in GAP, Magma, Sage, TeX

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

G:=Group("Dic3.M4(2)");
// GroupNames label

G:=SmallGroup(192,278);
// by ID

G=gap.SmallGroup(192,278);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,758,219,58,136,6278]);
// 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,b*d=d*b,d*c*d=a^3*c^5>;
// generators/relations

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