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## G = Q8×C3⋊S3order 144 = 24·32

### Direct product of Q8 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Q8×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — Q8×C3⋊S3
 Lower central C32 — C3×C6 — Q8×C3⋊S3
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C3⋊S3
G = < a,b,c,d,e | a4=c3=d3=e2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 322 in 114 conjugacy classes, 49 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, Q8, C32, Dic3, C12, D6, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C3×Q8, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8, C324Q8, C4×C3⋊S3, Q8×C32, Q8×C3⋊S3
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, C22×S3, C2×C3⋊S3, S3×Q8, C22×C3⋊S3, Q8×C3⋊S3

Character table of Q8×C3⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L size 1 1 9 9 2 2 2 2 2 2 2 18 18 18 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ9 2 2 0 0 -1 -1 -1 2 2 -2 -2 0 0 0 2 -1 -1 -1 2 -1 1 1 1 -2 1 -2 1 1 -1 -1 orthogonal lifted from D6 ρ10 2 2 0 0 -1 -1 2 -1 -2 2 -2 0 0 0 -1 2 -1 -1 1 1 -1 1 -2 1 1 -1 -1 2 1 -2 orthogonal lifted from D6 ρ11 2 2 0 0 -1 -1 2 -1 -2 -2 2 0 0 0 -1 2 -1 -1 1 1 1 -1 2 -1 -1 1 1 -2 1 -2 orthogonal lifted from D6 ρ12 2 2 0 0 -1 2 -1 -1 2 -2 -2 0 0 0 -1 -1 -1 2 -1 -1 -2 -2 1 1 1 1 1 1 2 -1 orthogonal lifted from D6 ρ13 2 2 0 0 2 -1 -1 -1 -2 -2 2 0 0 0 -1 -1 2 -1 1 -2 1 -1 -1 -1 2 1 -2 1 1 1 orthogonal lifted from D6 ρ14 2 2 0 0 -1 -1 -1 2 -2 2 -2 0 0 0 2 -1 -1 -1 -2 1 -1 1 1 -2 1 2 -1 -1 1 1 orthogonal lifted from D6 ρ15 2 2 0 0 -1 2 -1 -1 -2 2 -2 0 0 0 -1 -1 -1 2 1 1 2 -2 1 1 1 -1 -1 -1 -2 1 orthogonal lifted from D6 ρ16 2 2 0 0 -1 -1 2 -1 2 -2 -2 0 0 0 -1 2 -1 -1 -1 -1 1 1 -2 1 1 1 1 -2 -1 2 orthogonal lifted from D6 ρ17 2 2 0 0 -1 -1 2 -1 2 2 2 0 0 0 -1 2 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 -1 2 -1 2 orthogonal lifted from S3 ρ18 2 2 0 0 2 -1 -1 -1 2 2 2 0 0 0 -1 -1 2 -1 -1 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 0 0 -1 -1 -1 2 2 2 2 0 0 0 2 -1 -1 -1 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ20 2 2 0 0 -1 2 -1 -1 2 2 2 0 0 0 -1 -1 -1 2 -1 -1 2 2 -1 -1 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ21 2 2 0 0 2 -1 -1 -1 -2 2 -2 0 0 0 -1 -1 2 -1 1 -2 -1 1 1 1 -2 -1 2 -1 1 1 orthogonal lifted from D6 ρ22 2 2 0 0 -1 -1 -1 2 -2 -2 2 0 0 0 2 -1 -1 -1 -2 1 1 -1 -1 2 -1 -2 1 1 1 1 orthogonal lifted from D6 ρ23 2 2 0 0 -1 2 -1 -1 -2 -2 2 0 0 0 -1 -1 -1 2 1 1 -2 2 -1 -1 -1 1 1 1 -2 1 orthogonal lifted from D6 ρ24 2 2 0 0 2 -1 -1 -1 2 -2 -2 0 0 0 -1 -1 2 -1 -1 2 1 1 1 1 -2 1 -2 1 -1 -1 orthogonal lifted from D6 ρ25 2 -2 -2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ26 2 -2 2 -2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ27 4 -4 0 0 -2 -2 4 -2 0 0 0 0 0 0 2 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ28 4 -4 0 0 -2 4 -2 -2 0 0 0 0 0 0 2 2 2 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ29 4 -4 0 0 -2 -2 -2 4 0 0 0 0 0 0 -4 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ30 4 -4 0 0 4 -2 -2 -2 0 0 0 0 0 0 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of Q8×C3⋊S3
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 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 43 69)(10 44 70)(11 41 71)(12 42 72)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 53 45)(18 54 46)(19 55 47)(20 56 48)(25 68 61)(26 65 62)(27 66 63)(28 67 64)
(1 7 40)(2 8 37)(3 5 38)(4 6 39)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 31 51)(14 32 52)(15 29 49)(16 30 50)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 33 58)(22 34 59)(23 35 60)(24 36 57)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 3)(2 4)(5 40)(6 37)(7 38)(8 39)(9 25)(10 26)(11 27)(12 28)(13 33)(14 34)(15 35)(16 36)(17 19)(18 20)(21 31)(22 32)(23 29)(24 30)(41 63)(42 64)(43 61)(44 62)(45 55)(46 56)(47 53)(48 54)(49 60)(50 57)(51 58)(52 59)(65 70)(66 71)(67 72)(68 69)

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

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

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

Q8×C3⋊S3 is a maximal subgroup of
C3⋊S3.5Q16  D12.9D6  Dic6.9D6  D12.15D6  C24.32D6  C24.35D6  D12.25D6  S32×Q8  D1215D6  C3272- 1+4  C3292- 1+4  Q8⋊He3⋊C2  C329(S3×Q8)
Q8×C3⋊S3 is a maximal quotient of
C62.231C23  C122Dic6  C62.233C23  C62.240C23  C12.31D12  C62.259C23  C62.261C23  C329(S3×Q8)

Matrix representation of Q8×C3⋊S3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 11 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 6 8 0 0 0 0 10 7
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,11,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,10,0,0,0,0,8,7],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

Q8×C3⋊S3 in GAP, Magma, Sage, TeX

Q_8\times C_3\rtimes S_3
% in TeX

G:=Group("Q8xC3:S3");
// GroupNames label

G:=SmallGroup(144,174);
// by ID

G=gap.SmallGroup(144,174);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,116,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^3=e^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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