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G = D6⋊C46C4order 192 = 26·3

6th semidirect product of D6⋊C4 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C46C4, D64(C4⋊C4), C6.68(C4×D4), C6.41C22≀C2, (C2×C12).251D4, (C22×C4).58D6, C2.4(C232D6), C22.27(S3×Q8), C2.7(D6⋊Q8), C2.4(D63Q8), (C22×S3).89D4, C22.112(S3×D4), C6.49(C22⋊Q8), (C22×S3).10Q8, C34(C23.8Q8), C6.C4240C2, (C2×Dic3).177D4, C2.6(D6.D4), C2.19(Dic35D4), C22.58(C4○D12), (S3×C23).90C22, C23.307(C22×S3), (C22×C6).353C23, (C22×C12).347C22, C22.29(Q83S3), C6.51(C22.D4), (C22×Dic3).59C22, (C2×C4⋊C4)⋊7S3, (C2×C4)⋊4(C4×S3), (C6×C4⋊C4)⋊24C2, C6.21(C2×C4⋊C4), C2.22(S3×C4⋊C4), (C2×C12)⋊21(C2×C4), C2.13(C4×C3⋊D4), (C2×C6).84(C2×Q8), (C2×D6⋊C4).12C2, (C2×Dic3)⋊7(C2×C4), (C2×C6).334(C2×D4), (S3×C22×C4).19C2, C22.138(S3×C2×C4), (C2×Dic3⋊C4)⋊13C2, C22.68(C2×C3⋊D4), (C2×C6).190(C4○D4), (C2×C4).169(C3⋊D4), (C22×S3).42(C2×C4), (C2×C6).121(C22×C4), SmallGroup(192,548)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D6⋊C46C4
C1C3C6C2×C6C22×C6S3×C23C2×D6⋊C4 — D6⋊C46C4
C3C2×C6 — D6⋊C46C4
C1C23C2×C4⋊C4

Generators and relations for D6⋊C46C4
 G = < a,b,c,d | a6=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 632 in 234 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×4], C3, C4 [×10], C22 [×7], C22 [×16], S3 [×4], C6 [×7], C2×C4 [×4], C2×C4 [×26], C23, C23 [×10], Dic3 [×5], C12 [×5], D6 [×4], D6 [×12], C2×C6 [×7], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×9], C24, C4×S3 [×8], C2×Dic3 [×4], C2×Dic3 [×7], C2×C12 [×4], C2×C12 [×7], C22×S3 [×6], C22×S3 [×4], C22×C6, C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C23×C4, Dic3⋊C4 [×2], D6⋊C4 [×4], D6⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×6], C22×Dic3 [×3], C22×C12 [×3], S3×C23, C23.8Q8, C6.C42 [×2], C2×Dic3⋊C4, C2×D6⋊C4 [×2], C6×C4⋊C4, S3×C22×C4, D6⋊C46C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×6], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, S3×C2×C4, C4○D12, S3×D4 [×2], S3×Q8, Q83S3, C2×C3⋊D4, C23.8Q8, S3×C4⋊C4, Dic35D4, D6.D4, D6⋊Q8, C4×C3⋊D4, C232D6, D63Q8, D6⋊C46C4

Smallest permutation representation of D6⋊C46C4
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 27)(2 26)(3 25)(4 30)(5 29)(6 28)(7 83)(8 82)(9 81)(10 80)(11 79)(12 84)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)(31 52)(32 51)(33 50)(34 49)(35 54)(36 53)(37 46)(38 45)(39 44)(40 43)(41 48)(42 47)(55 73)(56 78)(57 77)(58 76)(59 75)(60 74)(61 67)(62 72)(63 71)(64 70)(65 69)(66 68)(85 91)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 64 16 56)(2 65 17 57)(3 66 18 58)(4 61 13 59)(5 62 14 60)(6 63 15 55)(7 51 91 43)(8 52 92 44)(9 53 93 45)(10 54 94 46)(11 49 95 47)(12 50 96 48)(19 75 27 67)(20 76 28 68)(21 77 29 69)(22 78 30 70)(23 73 25 71)(24 74 26 72)(31 87 39 79)(32 88 40 80)(33 89 41 81)(34 90 42 82)(35 85 37 83)(36 86 38 84)
(1 44 20 32)(2 45 21 33)(3 46 22 34)(4 47 23 35)(5 48 24 36)(6 43 19 31)(7 67 87 55)(8 68 88 56)(9 69 89 57)(10 70 90 58)(11 71 85 59)(12 72 86 60)(13 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(61 95 73 83)(62 96 74 84)(63 91 75 79)(64 92 76 80)(65 93 77 81)(66 94 78 82)

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P6A···6G12A···12L
order12···22222344444444444444446···612···12
size11···166662222244446666121212122···24···4

48 irreducible representations

dim11111112222222222444
type++++++++++-++-+
imageC1C2C2C2C2C2C4S3D4D4D4Q8D6C4○D4C4×S3C3⋊D4C4○D12S3×D4S3×Q8Q83S3
kernelD6⋊C46C4C6.C42C2×Dic3⋊C4C2×D6⋊C4C6×C4⋊C4S3×C22×C4D6⋊C4C2×C4⋊C4C2×Dic3C2×C12C22×S3C22×S3C22×C4C2×C6C2×C4C2×C4C22C22C22C22
# reps12121181222234444211

Matrix representation of D6⋊C46C4 in GL5(𝔽13)

10000
001200
01100
000120
000012
,
10000
00100
01000
000120
00011
,
120000
011900
04200
0001211
00011
,
50000
05000
00500
00080
00055

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,1],[12,0,0,0,0,0,11,4,0,0,0,9,2,0,0,0,0,0,12,1,0,0,0,11,1],[5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,8,5,0,0,0,0,5] >;

D6⋊C46C4 in GAP, Magma, Sage, TeX

D_6\rtimes C_4\rtimes_6C_4
% in TeX

G:=Group("D6:C4:6C4");
// GroupNames label

G:=SmallGroup(192,548);
// by ID

G=gap.SmallGroup(192,548);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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