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## G = Q8×D12order 192 = 26·3

### Direct product of Q8 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Q8×D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — Q8×D12
 Lower central C3 — C2×C6 — Q8×D12
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q8×D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 680 in 280 conjugacy classes, 123 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, S3×Q8, C6×Q8, D4×Q8, C122Q8, C4×D12, C4.D12, Q8×C12, C2×S3×Q8, Q8×D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, D12, C22×S3, C22×D4, C22×Q8, 2- 1+4, C2×D12, S3×Q8, S3×C23, D4×Q8, C22×D12, C2×S3×Q8, Q8○D12, Q8×D12

Smallest permutation representation of Q8×D12
On 96 points
Generators in S96
(1 22 65 95)(2 23 66 96)(3 24 67 85)(4 13 68 86)(5 14 69 87)(6 15 70 88)(7 16 71 89)(8 17 72 90)(9 18 61 91)(10 19 62 92)(11 20 63 93)(12 21 64 94)(25 74 56 39)(26 75 57 40)(27 76 58 41)(28 77 59 42)(29 78 60 43)(30 79 49 44)(31 80 50 45)(32 81 51 46)(33 82 52 47)(34 83 53 48)(35 84 54 37)(36 73 55 38)
(1 43 65 78)(2 44 66 79)(3 45 67 80)(4 46 68 81)(5 47 69 82)(6 48 70 83)(7 37 71 84)(8 38 72 73)(9 39 61 74)(10 40 62 75)(11 41 63 76)(12 42 64 77)(13 51 86 32)(14 52 87 33)(15 53 88 34)(16 54 89 35)(17 55 90 36)(18 56 91 25)(19 57 92 26)(20 58 93 27)(21 59 94 28)(22 60 95 29)(23 49 96 30)(24 50 85 31)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)(25 35)(26 34)(27 33)(28 32)(29 31)(37 39)(40 48)(41 47)(42 46)(43 45)(50 60)(51 59)(52 58)(53 57)(54 56)(61 71)(62 70)(63 69)(64 68)(65 67)(74 84)(75 83)(76 82)(77 81)(78 80)(85 95)(86 94)(87 93)(88 92)(89 91)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4H 4I 4J 4K 4L ··· 4Q 6A 6B 6C 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 ··· 4 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 6 6 6 6 2 2 ··· 2 4 4 4 12 ··· 12 2 2 2 2 2 2 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 D6 D12 2- 1+4 S3×Q8 Q8○D12 kernel Q8×D12 C12⋊2Q8 C4×D12 C4.D12 Q8×C12 C2×S3×Q8 C4×Q8 D12 C3×Q8 C42 C4⋊C4 C2×Q8 Q8 C6 C4 C2 # reps 1 3 3 6 1 2 1 4 4 3 3 1 8 1 2 2

Matrix representation of Q8×D12 in GL6(𝔽13)

 1 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 0 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 9 10 0 0 0 0 10 4
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,10,0,0,0,0,10,4],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Q8×D12 in GAP, Magma, Sage, TeX

Q_8\times D_{12}
% in TeX

G:=Group("Q8xD12");
// GroupNames label

G:=SmallGroup(192,1134);
// by ID

G=gap.SmallGroup(192,1134);
# by ID

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

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

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