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G = Q8×C9⋊C6  order 432 = 24·33

Direct product of Q8 and C9⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: Q8×C9⋊C6, Dic18⋊3C6, D9⋊(C3×Q8), C9⋊2(C6×Q8), (Q8×D9)⋊1C3, (Q8×C9)⋊2C6, C36.6(C2×C6), C32.(S3×Q8), (C4×D9).1C6, C12.23(S3×C6), D18.5(C2×C6), (C3×C12).30D6, C36.C6⋊3C2, C9⋊C12.4C22, C18.7(C22×C6), Dic9.4(C2×C6), (Q8×C32).13S3, 3- 1+2⋊2(C2×Q8), (Q8×3- 1+2)⋊2C2, (C2×3- 1+2).7C23, (C4×3- 1+2).6C22, C4.6(C2×C9⋊C6), C3.3(C3×S3×Q8), C6.41(S3×C2×C6), (C4×C9⋊C6).1C2, C2.8(C22×C9⋊C6), (C2×C9⋊C6).5C22, (C3×Q8).31(C3×S3), (C3×C6).33(C22×S3), SmallGroup(432,370)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — Q8×C9⋊C6
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C4×C9⋊C6 — Q8×C9⋊C6
Lower central
C9 — C18 — Q8×C9⋊C6
Upper central
C1 — C2 — Q8

Generators and relations for Q8×C9⋊C6
 G = < a,b,c,d | a4=c9=d6=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 398 in 120 conjugacy classes, 56 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, D9, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, 3- 1+2, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, C9⋊C6, C2×3- 1+2, Dic18, C4×D9, Q8×C9, Q8×C9, C3×Dic6, S3×C12, Q8×C32, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, Q8×D9, C3×S3×Q8, C36.C6, C4×C9⋊C6, Q8×3- 1+2, Q8×C9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, C3×Q8, C22×S3, C22×C6, S3×C6, S3×Q8, C6×Q8, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C3×S3×Q8, C22×C9⋊C6, Q8×C9⋊C6

Smallest permutation representation of Q8×C9⋊C6
►On 72 points
Generators in S72
(1 29 11 20)(2 30 12 21)(3 31 13 22)(4 32 14 23)(5 33 15 24)(6 34 16 25)(7 35 17 26)(8 36 18 27)(9 28 10 19)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)

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

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

50 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D6E6F6G9A9B9C12A12B12C12D···12I12J···12O18A18B18C36A···36I
order1222333444444666666699912121212···1212···1218181836···36
size119923322218181823399996664446···618···1866612···12

50 irreducible representations

dim11111111122222224466
type++++-+-+-++
imageC1C2C2C2C3C6C6C6Q8×C9⋊C6S3Q8D6C3×S3C3×Q8S3×C6S3×Q8C3×S3×Q8C9⋊C6C2×C9⋊C6
kernelQ8×C9⋊C6C36.C6C4×C9⋊C6Q8×3- 1+2Q8×D9Dic18C4×D9Q8×C9C1Q8×C32C9⋊C6C3×C12C3×Q8D9C12C32C3Q8C4
# reps1331266211232461213

Matrix representation of Q8×C9⋊C6 ►in GL10(𝔽37)

0010000000
0001000000
36000000000
03600000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
110270000000
011027000000
270260000000
027026000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
0100000000
363600000000
0001000000
003636000000
0000000100
000000363600
0000000001
000000003636
0000100000
0000010000
,
36000000000
1100000000
00360000000
0011000000
0000100000
000036360000
0000000010
000000003636
000000363600
0000000100

G:=sub<GL(10,GF(37))| [0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[11,0,27,0,0,0,0,0,0,0,0,11,0,27,0,0,0,0,0,0,27,0,26,0,0,0,0,0,0,0,0,27,0,26,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,36,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0],[36,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0] >;
 

Q8×C9⋊C6 in GAP, Magma, Sage, TeX 

Q_8\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("Q8xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(432,370);
 
// by ID
 
G=gap.SmallGroup(432,370);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,303,142,10085,1034,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^9=d^6=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^2>;
 
// generators/relations
 

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