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G = Q8x3- 1+2order 216 = 23·33

Direct product of Q8 and 3- 1+2

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

Aliases: Q8x3- 1+2, C36.3C6, C6.17C62, C9:2(C3xQ8), (Q8xC9):3C3, C12.8(C3xC6), C18.8(C2xC6), (C3xC12).5C6, C32.(C3xQ8), C3.3(Q8xC32), (Q8xC32).3C3, (C3xQ8).8C32, C4.(C2x3- 1+2), (C4x3- 1+2).3C2, C2.3(C22x3- 1+2), (C2x3- 1+2).8C22, (C3xC6).16(C2xC6), SmallGroup(216,81)

Series: Derived Chief Lower central Upper central

C1C6 — Q8x3- 1+2
C1C3C6C3xC6C2x3- 1+2C4x3- 1+2 — Q8x3- 1+2
C1C6 — Q8x3- 1+2
C1C6 — Q8x3- 1+2

Generators and relations for Q8x3- 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 >

Subgroups: 60 in 48 conjugacy classes, 42 normal (12 characteristic)
Quotients: C1, C2, C3, C22, C6, Q8, C32, C2xC6, C3xC6, C3xQ8, 3- 1+2, C62, C2x3- 1+2, Q8xC32, C22x3- 1+2, Q8x3- 1+2
3C3
3C6
3C12
3C12
3C12
3C3xQ8

Smallest permutation representation of Q8x3- 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)]])

Q8x3- 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++-
imageC1C2C3C3C6C6Q8C3xQ8C3xQ83- 1+2C2x3- 1+2Q8x3- 1+2
kernelQ8x3- 1+2C4x3- 1+2Q8xC9Q8xC32C36C3xC123- 1+2C9C32Q8C4C1
# reps1362186162262

Matrix representation of Q8x3- 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] >;

Q8x3- 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 Q8x3- 1+2 in TeX

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