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G = Q8×C9⋊C6order 432 = 24·33

Direct product of Q8 and C9⋊C6

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 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 12C 12D ··· 12I 12J ··· 12O 18A 18B 18C 36A ··· 36I order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 9 9 9 12 12 12 12 ··· 12 12 ··· 12 18 18 18 36 ··· 36 size 1 1 9 9 2 3 3 2 2 2 18 18 18 2 3 3 9 9 9 9 6 6 6 4 4 4 6 ··· 6 18 ··· 18 6 6 6 12 ··· 12

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 4 4 6 6 type + + + + - + - + - + + image C1 C2 C2 C2 C3 C6 C6 C6 Q8×C9⋊C6 S3 Q8 D6 C3×S3 C3×Q8 S3×C6 S3×Q8 C3×S3×Q8 C9⋊C6 C2×C9⋊C6 kernel Q8×C9⋊C6 C36.C6 C4×C9⋊C6 Q8×3- 1+2 Q8×D9 Dic18 C4×D9 Q8×C9 C1 Q8×C32 C9⋊C6 C3×C12 C3×Q8 D9 C12 C32 C3 Q8 C4 # reps 1 3 3 1 2 6 6 2 1 1 2 3 2 4 6 1 2 1 3

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

 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 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 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 36 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 1 0 0

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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