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## G = Q8×3- 1+2order 216 = 23·33

### Direct product of Q8 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Q8×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C4×3- 1+2 — Q8×3- 1+2
 Lower central C1 — C6 — Q8×3- 1+2
 Upper central C1 — C6 — Q8×3- 1+2

Generators and relations for Q8×3- 1+2
G = < a,b,c,d | a4=c9=d3=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of Q8×3- 1+2
On 72 points
Generators in S72
(1 37 19 34)(2 38 20 35)(3 39 21 36)(4 40 22 28)(5 41 23 29)(6 42 24 30)(7 43 25 31)(8 44 26 32)(9 45 27 33)(10 53 71 56)(11 54 72 57)(12 46 64 58)(13 47 65 59)(14 48 66 60)(15 49 67 61)(16 50 68 62)(17 51 69 63)(18 52 70 55)
(1 55 19 52)(2 56 20 53)(3 57 21 54)(4 58 22 46)(5 59 23 47)(6 60 24 48)(7 61 25 49)(8 62 26 50)(9 63 27 51)(10 38 71 35)(11 39 72 36)(12 40 64 28)(13 41 65 29)(14 42 66 30)(15 43 67 31)(16 44 68 32)(17 45 69 33)(18 37 70 34)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(20 26 23)(21 24 27)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(56 62 59)(57 60 63)(65 71 68)(66 69 72)

G:=sub<Sym(72)| (1,37,19,34)(2,38,20,35)(3,39,21,36)(4,40,22,28)(5,41,23,29)(6,42,24,30)(7,43,25,31)(8,44,26,32)(9,45,27,33)(10,53,71,56)(11,54,72,57)(12,46,64,58)(13,47,65,59)(14,48,66,60)(15,49,67,61)(16,50,68,62)(17,51,69,63)(18,52,70,55), (1,55,19,52)(2,56,20,53)(3,57,21,54)(4,58,22,46)(5,59,23,47)(6,60,24,48)(7,61,25,49)(8,62,26,50)(9,63,27,51)(10,38,71,35)(11,39,72,36)(12,40,64,28)(13,41,65,29)(14,42,66,30)(15,43,67,31)(16,44,68,32)(17,45,69,33)(18,37,70,34), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(56,62,59)(57,60,63)(65,71,68)(66,69,72)>;

G:=Group( (1,37,19,34)(2,38,20,35)(3,39,21,36)(4,40,22,28)(5,41,23,29)(6,42,24,30)(7,43,25,31)(8,44,26,32)(9,45,27,33)(10,53,71,56)(11,54,72,57)(12,46,64,58)(13,47,65,59)(14,48,66,60)(15,49,67,61)(16,50,68,62)(17,51,69,63)(18,52,70,55), (1,55,19,52)(2,56,20,53)(3,57,21,54)(4,58,22,46)(5,59,23,47)(6,60,24,48)(7,61,25,49)(8,62,26,50)(9,63,27,51)(10,38,71,35)(11,39,72,36)(12,40,64,28)(13,41,65,29)(14,42,66,30)(15,43,67,31)(16,44,68,32)(17,45,69,33)(18,37,70,34), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(56,62,59)(57,60,63)(65,71,68)(66,69,72) );

G=PermutationGroup([[(1,37,19,34),(2,38,20,35),(3,39,21,36),(4,40,22,28),(5,41,23,29),(6,42,24,30),(7,43,25,31),(8,44,26,32),(9,45,27,33),(10,53,71,56),(11,54,72,57),(12,46,64,58),(13,47,65,59),(14,48,66,60),(15,49,67,61),(16,50,68,62),(17,51,69,63),(18,52,70,55)], [(1,55,19,52),(2,56,20,53),(3,57,21,54),(4,58,22,46),(5,59,23,47),(6,60,24,48),(7,61,25,49),(8,62,26,50),(9,63,27,51),(10,38,71,35),(11,39,72,36),(12,40,64,28),(13,41,65,29),(14,42,66,30),(15,43,67,31),(16,44,68,32),(17,45,69,33),(18,37,70,34)], [(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)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(20,26,23),(21,24,27),(29,35,32),(30,33,36),(38,44,41),(39,42,45),(47,53,50),(48,51,54),(56,62,59),(57,60,63),(65,71,68),(66,69,72)]])

Q8×3- 1+2 is a maximal subgroup of   Dic18.C6  D36.C6  D363C6

55 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 9A ··· 9F 12A ··· 12F 12G ··· 12L 18A ··· 18F 36A ··· 36R order 1 2 3 3 3 3 4 4 4 6 6 6 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 3 3 2 2 2 1 1 3 3 3 ··· 3 2 ··· 2 6 ··· 6 3 ··· 3 6 ··· 6

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 type + + - image C1 C2 C3 C3 C6 C6 Q8 C3×Q8 C3×Q8 3- 1+2 C2×3- 1+2 Q8×3- 1+2 kernel Q8×3- 1+2 C4×3- 1+2 Q8×C9 Q8×C32 C36 C3×C12 3- 1+2 C9 C32 Q8 C4 C1 # reps 1 3 6 2 18 6 1 6 2 2 6 2

Matrix representation of Q8×3- 1+2 in GL5(𝔽37)

 1 31 0 0 0 25 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 31 0 0 0 0 35 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 27 0 0 0 0 10 0 0 25 36 36
,
 26 0 0 0 0 0 26 0 0 0 0 0 1 27 1 0 0 0 10 0 0 0 0 0 26

G:=sub<GL(5,GF(37))| [1,25,0,0,0,31,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[31,35,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,25,0,0,0,0,36,0,0,27,10,36],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,27,10,0,0,0,1,0,26] >;

Q8×3- 1+2 in GAP, Magma, Sage, TeX

Q_8\times 3_-^{1+2}
% in TeX

G:=Group("Q8xES-(3,1)");
// GroupNames label

G:=SmallGroup(216,81);
// by ID

G=gap.SmallGroup(216,81);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,457,223,338,519]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^9=d^3=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^-1=c^4>;
// generators/relations

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