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## G = Q8×C9order 72 = 23·32

### Direct product of C9 and Q8

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

Aliases: Q8×C9, C4.C18, C36.3C2, C12.4C6, C18.7C22, C3.(C3×Q8), C18(C3×Q8), C6.7(C2×C6), C2.2(C2×C18), (C3×Q8).2C3, SmallGroup(72,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C9
 Chief series C1 — C3 — C6 — C18 — C36 — Q8×C9
 Lower central C1 — C2 — Q8×C9
 Upper central C1 — C18 — Q8×C9

Generators and relations for Q8×C9
G = < a,b,c | a9=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Q8×C9
Regular action on 72 points
Generators in S72
(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)
(1 40 22 34)(2 41 23 35)(3 42 24 36)(4 43 25 28)(5 44 26 29)(6 45 27 30)(7 37 19 31)(8 38 20 32)(9 39 21 33)(10 48 66 63)(11 49 67 55)(12 50 68 56)(13 51 69 57)(14 52 70 58)(15 53 71 59)(16 54 72 60)(17 46 64 61)(18 47 65 62)
(1 58 22 52)(2 59 23 53)(3 60 24 54)(4 61 25 46)(5 62 26 47)(6 63 27 48)(7 55 19 49)(8 56 20 50)(9 57 21 51)(10 45 66 30)(11 37 67 31)(12 38 68 32)(13 39 69 33)(14 40 70 34)(15 41 71 35)(16 42 72 36)(17 43 64 28)(18 44 65 29)

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

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

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

Q8×C9 is a maximal subgroup of   C9⋊Q16  Q82D9  Q83D9  Q8⋊C27  C18.A4  2+ 1+4⋊C9

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + - image C1 C2 C3 C6 C9 C18 Q8 C3×Q8 Q8×C9 kernel Q8×C9 C36 C3×Q8 C12 Q8 C4 C9 C3 C1 # reps 1 3 2 6 6 18 1 2 6

Matrix representation of Q8×C9 in GL2(𝔽19) generated by

 9 0 0 9
,
 16 8 13 3
,
 0 9 2 0
G:=sub<GL(2,GF(19))| [9,0,0,9],[16,13,8,3],[0,2,9,0] >;

Q8×C9 in GAP, Magma, Sage, TeX

Q_8\times C_9
% in TeX

G:=Group("Q8xC9");
// GroupNames label

G:=SmallGroup(72,11);
// by ID

G=gap.SmallGroup(72,11);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,60,141,66,102]);
// Polycyclic

G:=Group<a,b,c|a^9=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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