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G = D46D12order 192 = 26·3

2nd semidirect product of D4 and D12 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D46D12, C42.112D6, C6.612- 1+4, (C3×D4)⋊11D4, (C4×D4)⋊17S3, (D4×C12)⋊19C2, (C4×D12)⋊31C2, C33(D46D4), C4⋊C4.284D6, C12.55(C2×D4), C4.23(C2×D12), C1214(C4○D4), C45(D42S3), C127D410C2, C4.D1215C2, C122Q825C2, (C2×D4).249D6, (C2×C6).99C24, D6⋊C4.5C22, C6.17(C22×D4), C22.2(C2×D12), C22⋊C4.113D6, C2.18(Q8○D12), C2.19(C22×D12), (C22×C4).227D6, (C4×C12).155C22, (C2×C12).160C23, (C6×D4).260C22, C23.21D66C2, C4⋊Dic3.39C22, (C2×D12).212C22, (C22×S3).34C23, C23.183(C22×S3), (C22×C6).169C23, (C22×C12).81C22, C22.124(S3×C23), (C2×Dic6).144C22, (C2×Dic3).206C23, (C22×Dic3).97C22, (C2×C6).2(C2×D4), C6.74(C2×C4○D4), (C2×D42S3)⋊4C2, (C2×C4⋊Dic3)⋊25C2, (S3×C2×C4).65C22, C2.22(C2×D42S3), (C3×C4⋊C4).329C22, (C2×C4).732(C22×S3), (C2×C3⋊D4).15C22, (C3×C22⋊C4).106C22, SmallGroup(192,1114)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D46D12
C1C3C6C2×C6C22×S3S3×C2×C4C2×D42S3 — D46D12
C3C2×C6 — D46D12
C1C22C4×D4

Generators and relations for D46D12
 G = < a,b,c,d | a4=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 728 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], Dic3 [×6], C12 [×4], C12 [×3], D6 [×6], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C4⋊Dic3, C4⋊Dic3 [×8], D6⋊C4 [×6], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, D42S3 [×8], C22×Dic3 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, D46D4, C122Q8, C4×D12, C23.21D6 [×4], C4.D12 [×2], C2×C4⋊Dic3 [×2], C127D4 [×2], D4×C12, C2×D42S3 [×2], D46D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2- 1+4, C2×D12 [×6], D42S3 [×2], S3×C23, D46D4, C22×D12, C2×D42S3, Q8○D12, D46D12

Smallest permutation representation of D46D12
On 96 points
Generators in S96
(1 83 29 72)(2 84 30 61)(3 73 31 62)(4 74 32 63)(5 75 33 64)(6 76 34 65)(7 77 35 66)(8 78 36 67)(9 79 25 68)(10 80 26 69)(11 81 27 70)(12 82 28 71)(13 57 88 46)(14 58 89 47)(15 59 90 48)(16 60 91 37)(17 49 92 38)(18 50 93 39)(19 51 94 40)(20 52 95 41)(21 53 96 42)(22 54 85 43)(23 55 86 44)(24 56 87 45)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 85)(9 86)(10 87)(11 88)(12 89)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 91)(14 90)(15 89)(16 88)(17 87)(18 86)(19 85)(20 96)(21 95)(22 94)(23 93)(24 92)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 60)(47 59)(48 58)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O6A6B6C6D6E6F6G12A12B12C12D12E···12L
order1222222222344444444444444466666661212121212···12
size11112222121222222444666612121212222444422224···4

45 irreducible representations

dim111111111222222222444
type+++++++++++++++++---
imageC1C2C2C2C2C2C2C2C2S3D4D6D6D6D6D6C4○D4D122- 1+4D42S3Q8○D12
kernelD46D12C122Q8C4×D12C23.21D6C4.D12C2×C4⋊Dic3C127D4D4×C12C2×D42S3C4×D4C3×D4C42C22⋊C4C4⋊C4C22×C4C2×D4C12D4C6C4C2
# reps111422212141212148122

Matrix representation of D46D12 in GL4(𝔽13) generated by

5000
0800
00120
00012
,
0800
5000
0010
0001
,
1000
0100
00310
0036
,
1000
01200
00310
00710
G:=sub<GL(4,GF(13))| [5,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[0,5,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,3,0,0,10,6],[1,0,0,0,0,12,0,0,0,0,3,7,0,0,10,10] >;

D46D12 in GAP, Magma, Sage, TeX

D_4\rtimes_6D_{12}
% in TeX

G:=Group("D4:6D12");
// GroupNames label

G:=SmallGroup(192,1114);
// by ID

G=gap.SmallGroup(192,1114);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,675,80,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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