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G = Q87D12order 192 = 26·3

2nd semidirect product of Q8 and D12 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q87D12, C42.129D6, C6.1102+ 1+4, (C3×Q8)⋊11D4, (C4×Q8)⋊14S3, (C4×D12)⋊39C2, C4⋊C4.296D6, (Q8×C12)⋊12C2, C32(Q86D4), C12.58(C2×D4), C4.26(C2×D12), C12⋊D418C2, C1217(C4○D4), C4⋊D1213C2, C43(Q83S3), (C2×Q8).229D6, C6.20(C22×D4), C2.22(D4○D12), (C2×C6).121C24, C2.22(C22×D12), (C2×C12).170C23, (C4×C12).173C22, D6⋊C4.101C22, (C2×D12).28C22, (C6×Q8).221C22, (C22×S3).46C23, C4⋊Dic3.399C22, C22.142(S3×C23), (C2×Dic3).215C23, (C2×Q83S3)⋊4C2, C6.112(C2×C4○D4), (S3×C2×C4).73C22, C2.11(C2×Q83S3), (C3×C4⋊C4).349C22, (C2×C4).734(C22×S3), SmallGroup(192,1136)

Series: Derived Chief Lower central Upper central

C1C2×C6 — Q87D12
C1C3C6C2×C6C22×S3S3×C2×C4C2×Q83S3 — Q87D12
C3C2×C6 — Q87D12
C1C22C4×Q8

Generators and relations for Q87D12
 G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 936 in 312 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×8], C4 [×5], C22, C22 [×18], S3 [×6], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×24], Q8 [×4], C23 [×6], Dic3 [×2], C12 [×8], C12 [×3], D6 [×18], C2×C6, C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4, C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], C4×S3 [×12], D12 [×24], C2×Dic3 [×2], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3 [×6], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C41D4 [×3], C2×C4○D4 [×2], C4⋊Dic3, D6⋊C4 [×6], C4×C12 [×3], C3×C4⋊C4 [×3], S3×C2×C4 [×6], C2×D12 [×15], Q83S3 [×8], C6×Q8, Q86D4, C4×D12 [×3], C4⋊D12 [×3], C12⋊D4 [×6], Q8×C12, C2×Q83S3 [×2], Q87D12
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], Q83S3 [×2], S3×C23, Q86D4, C22×D12, C2×Q83S3, D4○D12, Q87D12

Smallest permutation representation of Q87D12
On 96 points
Generators in S96
(1 35 42 50)(2 36 43 51)(3 25 44 52)(4 26 45 53)(5 27 46 54)(6 28 47 55)(7 29 48 56)(8 30 37 57)(9 31 38 58)(10 32 39 59)(11 33 40 60)(12 34 41 49)(13 92 62 74)(14 93 63 75)(15 94 64 76)(16 95 65 77)(17 96 66 78)(18 85 67 79)(19 86 68 80)(20 87 69 81)(21 88 70 82)(22 89 71 83)(23 90 72 84)(24 91 61 73)
(1 70 42 21)(2 71 43 22)(3 72 44 23)(4 61 45 24)(5 62 46 13)(6 63 47 14)(7 64 48 15)(8 65 37 16)(9 66 38 17)(10 67 39 18)(11 68 40 19)(12 69 41 20)(25 90 52 84)(26 91 53 73)(27 92 54 74)(28 93 55 75)(29 94 56 76)(30 95 57 77)(31 96 58 78)(32 85 59 79)(33 86 60 80)(34 87 49 81)(35 88 50 82)(36 89 51 83)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 60)(9 59)(10 58)(11 57)(12 56)(13 89)(14 88)(15 87)(16 86)(17 85)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D···2I 3 4A···4H4I4J4K4L4M4N4O6A6B6C12A12B12C12D12E···12P
order12222···234···444444446661212121212···12
size111112···1222···2444666622222224···4

45 irreducible representations

dim1111112222222444
type+++++++++++++++
imageC1C2C2C2C2C2S3D4D6D6D6C4○D4D122+ 1+4Q83S3D4○D12
kernelQ87D12C4×D12C4⋊D12C12⋊D4Q8×C12C2×Q83S3C4×Q8C3×Q8C42C4⋊C4C2×Q8C12Q8C6C4C2
# reps1336121433148122

Matrix representation of Q87D12 in GL4(𝔽13) generated by

1000
0100
00128
0031
,
12000
01200
00512
0008
,
31000
3600
00120
00012
,
10300
6300
0015
00012
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,3,0,0,8,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,12,8],[3,3,0,0,10,6,0,0,0,0,12,0,0,0,0,12],[10,6,0,0,3,3,0,0,0,0,1,0,0,0,5,12] >;

Q87D12 in GAP, Magma, Sage, TeX

Q_8\rtimes_7D_{12}
% in TeX

G:=Group("Q8:7D12");
// GroupNames label

G:=SmallGroup(192,1136);
// by ID

G=gap.SmallGroup(192,1136);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,184,675,80,6278]);
// Polycyclic

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

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