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G = M4(2)⋊Dic3order 192 = 26·3

2nd semidirect product of M4(2) and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊2Dic3, C12.9(C4⋊C4), (C2×C12).9Q8, C4.Dic35C4, (C2×C6).3C42, (C2×C4).4Dic6, C4.34(D6⋊C4), (C2×C12).468D4, (C2×C4).128D12, (C3×M4(2))⋊5C4, C23.48(C4×S3), C4.3(C4⋊Dic3), (C22×C4).77D6, C6.5(C4.D4), (C2×M4(2)).5S3, (C6×M4(2)).9C2, C22.3(C4×Dic3), C12.86(C22⋊C4), C32(C22.C42), C4.24(Dic3⋊C4), C22.41(D6⋊C4), C6.5(C4.10D4), (C22×Dic3).3C4, C2.2(C12.47D4), C2.2(C12.46D4), C4.17(C6.D4), C22.5(Dic3⋊C4), C2.13(C6.C42), C6.13(C2.C42), (C22×C12).123C22, (C2×C6).6(C4⋊C4), (C2×C4).20(C4×S3), (C2×C12).61(C2×C4), (C2×C4).21(C3⋊D4), (C2×C4⋊Dic3).28C2, (C22×C6).31(C2×C4), (C2×C4).14(C2×Dic3), (C2×C6).53(C22⋊C4), (C2×C4.Dic3).10C2, SmallGroup(192,113)

Series: Derived Chief Lower central Upper central

C1C2×C6 — M4(2)⋊Dic3
C1C3C6C2×C6C2×C12C22×C12C2×C4⋊Dic3 — M4(2)⋊Dic3
C3C6C2×C6 — M4(2)⋊Dic3
C1C22C22×C4C2×M4(2)

Generators and relations for M4(2)⋊Dic3
 G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 232 in 98 conjugacy classes, 51 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×2], C22 [×3], C22 [×2], C6 [×3], C6 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C23, Dic3 [×2], C12 [×4], C2×C6 [×3], C2×C6 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×4], C2×C12 [×6], C22×C6, C2×C4⋊C4, C2×M4(2), C2×M4(2), C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C4⋊Dic3 [×2], C2×C24, C3×M4(2) [×2], C3×M4(2), C22×Dic3 [×2], C22×C12, C22.C42, C2×C4.Dic3, C2×C4⋊Dic3, C6×M4(2), M4(2)⋊Dic3
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2.C42, C4.D4, C4.10D4, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C22.C42, C12.46D4, C12.47D4, C6.C42, M4(2)⋊Dic3

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A···24H
order122222344444444666668888888812121212121224···24
size1111222222212121212222444444121212122222444···4

42 irreducible representations

dim111111122222222224444
type++++++--+-++-+-
imageC1C2C2C2C4C4C4S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4C4×S3C4.D4C4.10D4C12.46D4C12.47D4
kernelM4(2)⋊Dic3C2×C4.Dic3C2×C4⋊Dic3C6×M4(2)C4.Dic3C3×M4(2)C22×Dic3C2×M4(2)C2×C12C2×C12M4(2)C22×C4C2×C4C2×C4C2×C4C2×C4C23C6C6C2C2
# reps111144413121222421122

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

100000
010000
0043132525
0060302325
0062634360
0072621330
,
100000
010000
001000
000100
001717720
00017072
,
1720000
100000
000100
00727200
0000721
0000720
,
0270000
2700000
0052300
00186800
002614505
0014615523

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,60,62,72,0,0,13,30,63,62,0,0,25,23,43,13,0,0,25,25,60,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,17,0,0,0,0,1,17,17,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,27,0,0,0,0,27,0,0,0,0,0,0,0,5,18,26,14,0,0,23,68,14,61,0,0,0,0,50,55,0,0,0,0,5,23] >;

M4(2)⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,113);
// by ID

G=gap.SmallGroup(192,113);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,1123,136,851,6278]);
// Polycyclic

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

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