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G = Q8×3- 1+2  order 216 = 23·33

Direct product of Q8 and 3- 1+2

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

Aliases: Q8×3- 1+2, C36.3C6, C6.17C62, C9⋊2(C3×Q8), (Q8×C9)⋊3C3, C12.8(C3×C6), C18.8(C2×C6), (C3×C12).5C6, C32.(C3×Q8), C3.3(Q8×C32), (Q8×C32).3C3, (C3×Q8).8C32, C4.(C2×3- 1+2), (C4×3- 1+2).3C2, C2.3(C22×3- 1+2), (C2×3- 1+2).8C22, (C3×C6).16(C2×C6), SmallGroup(216,81)

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 >

C1
C2
C3
3C3
C4
C4
C4
C6
3C6
C9
C9
C9
C32
Q8
C12
C12
C12
3C12
3C12
3C12
C3×C6
C18
C18
C18
3- 1+2
C3×Q8
3C3×Q8
C36
C36
C3×C12
C36
C36
C36
C3×C12
C36
C36
C36
C36
C3×C12
C2×3- 1+2
Q8×C9
Q8×C9
Q8×C32
Q8×C9
C4×3- 1+2
C4×3- 1+2
C4×3- 1+2
Q8×3- 1+2

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  D36⋊3C6

55 conjugacy classes

class 1  2 3A3B3C3D4A4B4C6A6B6C6D9A···9F12A···12F12G···12L18A···18F36A···36R
order12333344466669···912···1212···1218···1836···36
size11113322211333···32···26···63···36···6

55 irreducible representations

dim111111222336
type++-
imageC1C2C3C3C6C6Q8C3×Q8C3×Q83- 1+2C2×3- 1+2Q8×3- 1+2
kernelQ8×3- 1+2C4×3- 1+2Q8×C9Q8×C32C36C3×C123- 1+2C9C32Q8C4C1
# reps1362186162262

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

131000
2536000
00100
00010
00001
,
310000
356000
00100
00010
00001
,
10000
01000
001027
000010
00253636
,
260000
026000
001271
000100
000026

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
 

Export

Subgroup lattice of Q8×3- 1+2 in TeX

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