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G = D126Q8order 192 = 26·3

4th semidirect product of D12 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D126Q8, C12.18D8, C42.81D6, C4⋊Q85S3, C4⋊C4.83D6, C6.60(C2×D8), C4.12(S3×Q8), C35(D4⋊Q8), C12⋊C835C2, C12.39(C2×Q8), C4.16(D4⋊S3), (C4×D12).19C2, C6.Q1643C2, (C2×C12).157D4, C12.82(C4○D4), C6.D8.15C2, C6.76(C22⋊Q8), (C4×C12).134C22, (C2×C12).405C23, C2.13(D63Q8), C4.35(Q83S3), C6.97(C8.C22), (C2×D12).249C22, C4⋊Dic3.348C22, C2.18(Q8.11D6), (C3×C4⋊Q8)⋊5C2, C2.15(C2×D4⋊S3), (C2×C6).536(C2×D4), (C2×C3⋊C8).138C22, (C2×C4).189(C3⋊D4), (C3×C4⋊C4).130C22, (C2×C4).502(C22×S3), C22.208(C2×C3⋊D4), SmallGroup(192,646)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D126Q8
C1C3C6C12C2×C12C2×D12C4×D12 — D126Q8
C3C6C2×C12 — D126Q8
C1C22C42C4⋊Q8

Generators and relations for D126Q8
 G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 320 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], C23, Dic3, C12 [×2], C12 [×2], C12 [×3], D6 [×4], C2×C6, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C22×S3, D4⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, D4⋊Q8, C12⋊C8, C6.Q16 [×2], C6.D8 [×2], C4×D12, C3×C4⋊Q8, D126Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], D8 [×2], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×D8, C8.C22, D4⋊S3 [×2], S3×Q8, Q83S3, C2×C3⋊D4, D4⋊Q8, C2×D4⋊S3, Q8.11D6, D63Q8, D126Q8

Smallest permutation representation of D126Q8
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 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 26)(14 25)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 96)(48 95)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 84)(56 83)(57 82)(58 81)(59 80)(60 79)
(1 79 67 52)(2 74 68 59)(3 81 69 54)(4 76 70 49)(5 83 71 56)(6 78 72 51)(7 73 61 58)(8 80 62 53)(9 75 63 60)(10 82 64 55)(11 77 65 50)(12 84 66 57)(13 94 26 46)(14 89 27 41)(15 96 28 48)(16 91 29 43)(17 86 30 38)(18 93 31 45)(19 88 32 40)(20 95 33 47)(21 90 34 42)(22 85 35 37)(23 92 36 44)(24 87 25 39)
(1 90 67 42)(2 91 68 43)(3 92 69 44)(4 93 70 45)(5 94 71 46)(6 95 72 47)(7 96 61 48)(8 85 62 37)(9 86 63 38)(10 87 64 39)(11 88 65 40)(12 89 66 41)(13 56 26 83)(14 57 27 84)(15 58 28 73)(16 59 29 74)(17 60 30 75)(18 49 31 76)(19 50 32 77)(20 51 33 78)(21 52 34 79)(22 53 35 80)(23 54 36 81)(24 55 25 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), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,26)(14,25)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,96)(48,95)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79), (1,79,67,52)(2,74,68,59)(3,81,69,54)(4,76,70,49)(5,83,71,56)(6,78,72,51)(7,73,61,58)(8,80,62,53)(9,75,63,60)(10,82,64,55)(11,77,65,50)(12,84,66,57)(13,94,26,46)(14,89,27,41)(15,96,28,48)(16,91,29,43)(17,86,30,38)(18,93,31,45)(19,88,32,40)(20,95,33,47)(21,90,34,42)(22,85,35,37)(23,92,36,44)(24,87,25,39), (1,90,67,42)(2,91,68,43)(3,92,69,44)(4,93,70,45)(5,94,71,46)(6,95,72,47)(7,96,61,48)(8,85,62,37)(9,86,63,38)(10,87,64,39)(11,88,65,40)(12,89,66,41)(13,56,26,83)(14,57,27,84)(15,58,28,73)(16,59,29,74)(17,60,30,75)(18,49,31,76)(19,50,32,77)(20,51,33,78)(21,52,34,79)(22,53,35,80)(23,54,36,81)(24,55,25,82)>;

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C8A8B8C8D12A···12F12G12H12I12J
order1222223444444444666888812···1212121212
size11111212222224881212222121212124···48888

33 irreducible representations

dim1111112222222244444
type+++++++-++++-+-+
imageC1C2C2C2C2C2S3Q8D4D6D6D8C4○D4C3⋊D4C8.C22D4⋊S3S3×Q8Q83S3Q8.11D6
kernelD126Q8C12⋊C8C6.Q16C6.D8C4×D12C3×C4⋊Q8C4⋊Q8D12C2×C12C42C4⋊C4C12C12C2×C4C6C4C4C4C2
# reps1122111221242412112

Matrix representation of D126Q8 in GL6(𝔽73)

7200000
0720000
001100
0072000
0000723
0000481
,
100000
59720000
001100
0007200
0000723
000001
,
2130000
23520000
00431300
00603000
0000025
0000380
,
2700000
60460000
001000
000100
000010
000001

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,48,0,0,0,0,3,1],[1,59,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,3,1],[21,23,0,0,0,0,3,52,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,0,0,0,0,0,38,0,0,0,0,25,0],[27,60,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D126Q8 in GAP, Magma, Sage, TeX

D_{12}\rtimes_6Q_8
% in TeX

G:=Group("D12:6Q8");
// GroupNames label

G:=SmallGroup(192,646);
// by ID

G=gap.SmallGroup(192,646);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^7,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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