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## G = Dic3⋊4M4(2)  order 192 = 26·3

### 2nd semidirect product of Dic3 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3⋊4M4(2)
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C2×C4×Dic3 — Dic3⋊4M4(2)
 Lower central C3 — C2×C6 — Dic3⋊4M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for Dic34M4(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=c5 >

Subgroups: 248 in 126 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, 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), C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C4⋊C8, 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), Dic3⋊C8, C2×C4.Dic3, C2×C4×Dic3, C6×M4(2), Dic34M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C2×M4(2), Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C4⋊M4(2), S3×M4(2), C2×Dic3⋊C4, Dic34M4(2)

Smallest permutation representation of Dic34M4(2)
On 96 points
Generators in S96
(1 88 19 62 47 78)(2 81 20 63 48 79)(3 82 21 64 41 80)(4 83 22 57 42 73)(5 84 23 58 43 74)(6 85 24 59 44 75)(7 86 17 60 45 76)(8 87 18 61 46 77)(9 89 71 53 35 27)(10 90 72 54 36 28)(11 91 65 55 37 29)(12 92 66 56 38 30)(13 93 67 49 39 31)(14 94 68 50 40 32)(15 95 69 51 33 25)(16 96 70 52 34 26)
(1 30 62 66)(2 67 63 31)(3 32 64 68)(4 69 57 25)(5 26 58 70)(6 71 59 27)(7 28 60 72)(8 65 61 29)(9 75 53 24)(10 17 54 76)(11 77 55 18)(12 19 56 78)(13 79 49 20)(14 21 50 80)(15 73 51 22)(16 23 52 74)(33 83 95 42)(34 43 96 84)(35 85 89 44)(36 45 90 86)(37 87 91 46)(38 47 92 88)(39 81 93 48)(40 41 94 82)
(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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)(81 85)(83 87)(89 93)(91 95)

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

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

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

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 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 S3 D4 Q8 D6 D6 M4(2) Dic6 C4×S3 C3⋊D4 C4×S3 S3×M4(2) kernel Dic3⋊4M4(2) Dic3⋊C8 C2×C4.Dic3 C2×C4×Dic3 C6×M4(2) C4×Dic3 C22×Dic3 C2×M4(2) C2×C12 C2×C12 C2×C8 C22×C4 Dic3 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 4 1 2 2 2 1 8 4 2 4 2 4

Matrix representation of Dic34M4(2) in GL4(𝔽73) generated by

 0 72 0 0 1 1 0 0 0 0 72 0 0 0 0 72
,
 65 39 0 0 47 8 0 0 0 0 27 0 0 0 7 46
,
 43 13 0 0 60 30 0 0 0 0 28 3 0 0 46 45
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 30 72
G:=sub<GL(4,GF(73))| [0,1,0,0,72,1,0,0,0,0,72,0,0,0,0,72],[65,47,0,0,39,8,0,0,0,0,27,7,0,0,0,46],[43,60,0,0,13,30,0,0,0,0,28,46,0,0,3,45],[1,0,0,0,0,1,0,0,0,0,1,30,0,0,0,72] >;

Dic34M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_4M_4(2)
% in TeX

G:=Group("Dic3:4M4(2)");
// GroupNames label

G:=SmallGroup(192,677);
// by ID

G=gap.SmallGroup(192,677);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,387,58,136,6278]);
// 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=c^5>;
// generators/relations

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