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## G = Dic3⋊5M4(2)  order 192 = 26·3

### The semidirect product of Dic3 and M4(2) acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Dic3⋊5M4(2)
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C42 — Dic3⋊5M4(2)
 Lower central C3 — C6 — Dic3⋊5M4(2)
 Upper central C1 — C2×C4 — C8⋊C4

Generators and relations for Dic35M4(2)
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c5 >

Subgroups: 280 in 142 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C8⋊C4, C8⋊C4, C2×C42, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, S3×C2×C4, C4×M4(2), C42.S3, C8×Dic3, C3×C8⋊C4, S3×C42, C2×C8⋊S3, Dic35M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, M4(2), C22×C4, C4×S3, C22×S3, C2×C42, C2×M4(2), S3×C2×C4, C4×M4(2), S3×C42, S3×M4(2), Dic35M4(2)

Smallest permutation representation of Dic35M4(2)
On 96 points
Generators in S96
(1 62 41 23 74 51)(2 52 75 24 42 63)(3 64 43 17 76 53)(4 54 77 18 44 57)(5 58 45 19 78 55)(6 56 79 20 46 59)(7 60 47 21 80 49)(8 50 73 22 48 61)(9 40 91 69 84 31)(10 32 85 70 92 33)(11 34 93 71 86 25)(12 26 87 72 94 35)(13 36 95 65 88 27)(14 28 81 66 96 37)(15 38 89 67 82 29)(16 30 83 68 90 39)
(1 66 23 14)(2 67 24 15)(3 68 17 16)(4 69 18 9)(5 70 19 10)(6 71 20 11)(7 72 21 12)(8 65 22 13)(25 56 93 46)(26 49 94 47)(27 50 95 48)(28 51 96 41)(29 52 89 42)(30 53 90 43)(31 54 91 44)(32 55 92 45)(33 58 85 78)(34 59 86 79)(35 60 87 80)(36 61 88 73)(37 62 81 74)(38 63 82 75)(39 64 83 76)(40 57 84 77)
(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 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 34)(26 39)(27 36)(28 33)(29 38)(30 35)(31 40)(32 37)(41 78)(42 75)(43 80)(44 77)(45 74)(46 79)(47 76)(48 73)(49 64)(50 61)(51 58)(52 63)(53 60)(54 57)(55 62)(56 59)(66 70)(68 72)(81 92)(82 89)(83 94)(84 91)(85 96)(86 93)(87 90)(88 95)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 S3×M4(2) kernel Dic3⋊5M4(2) C42.S3 C8×Dic3 C3×C8⋊C4 S3×C42 C2×C8⋊S3 C8⋊S3 C4×Dic3 S3×C2×C4 C8⋊C4 C42 C2×C8 Dic3 C8 C2×C4 C2 # reps 1 1 2 1 1 2 16 4 4 1 1 2 8 8 4 4

Matrix representation of Dic35M4(2) in GL4(𝔽73) generated by

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

Dic35M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_5M_4(2)
% in TeX

G:=Group("Dic3:5M4(2)");
// GroupNames label

G:=SmallGroup(192,266);
// by ID

G=gap.SmallGroup(192,266);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,387,58,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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