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G = Q8×C3⋊S3order 144 = 24·32

Direct product of Q8 and C3⋊S3

direct product, metabelian, supersoluble, monomial, rational

Aliases: Q8×C3⋊S3, C12.25D6, C33(S3×Q8), (C3×Q8)⋊4S3, C328(C2×Q8), (Q8×C32)⋊5C2, C324Q87C2, C6.36(C22×S3), (C3×C6).35C23, (C3×C12).25C22, C3⋊Dic3.19C22, C4.6(C2×C3⋊S3), (C4×C3⋊S3).2C2, C2.8(C22×C3⋊S3), (C2×C3⋊S3).22C22, SmallGroup(144,174)

Series: Derived Chief Lower central Upper central

C1C3×C6 — Q8×C3⋊S3
C1C3C32C3×C6C2×C3⋊S3C4×C3⋊S3 — Q8×C3⋊S3
C32C3×C6 — Q8×C3⋊S3
C1C2Q8

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 [×2], C3 [×4], C4 [×3], C4 [×3], C22, S3 [×8], C6 [×4], C2×C4 [×3], Q8, Q8 [×3], C32, Dic3 [×12], C12 [×12], D6 [×4], C2×Q8, C3⋊S3 [×2], C3×C6, Dic6 [×12], C4×S3 [×12], C3×Q8 [×4], C3⋊Dic3 [×3], C3×C12 [×3], C2×C3⋊S3, S3×Q8 [×4], C324Q8 [×3], C4×C3⋊S3 [×3], Q8×C32, Q8×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], Q8 [×2], C23, D6 [×12], C2×Q8, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], S3×Q8 [×4], C22×C3⋊S3, Q8×C3⋊S3

Character table of Q8×C3⋊S3

 class 12A2B2C3A3B3C3D4A4B4C4D4E4F6A6B6C6D12A12B12C12D12E12F12G12H12I12J12K12L
 size 119922222221818182222444444444444
ρ1111111111111111111111111111111    trivial
ρ211-1-111111-1-111-1111111-1-1-1-1-1-1-1-111    linear of order 2
ρ311-1-11111111-1-1-11111111111111111    linear of order 2
ρ4111111111-1-1-1-11111111-1-1-1-1-1-1-1-111    linear of order 2
ρ511-1-11111-1-11-1111111-1-1-11111-1-1-1-1-1    linear of order 2
ρ611111111-11-1-11-11111-1-11-1-1-1-1111-1-1    linear of order 2
ρ711111111-1-111-1-11111-1-1-11111-1-1-1-1-1    linear of order 2
ρ811-1-11111-11-11-111111-1-11-1-1-1-1111-1-1    linear of order 2
ρ92200-1-1-122-2-20002-1-1-12-1111-21-211-1-1    orthogonal lifted from D6
ρ102200-1-12-1-22-2000-12-1-111-11-211-1-121-2    orthogonal lifted from D6
ρ112200-1-12-1-2-22000-12-1-1111-12-1-111-21-2    orthogonal lifted from D6
ρ122200-12-1-12-2-2000-1-1-12-1-1-2-21111112-1    orthogonal lifted from D6
ρ1322002-1-1-1-2-22000-1-12-11-21-1-1-121-2111    orthogonal lifted from D6
ρ142200-1-1-12-22-20002-1-1-1-21-111-212-1-111    orthogonal lifted from D6
ρ152200-12-1-1-22-2000-1-1-12112-2111-1-1-1-21    orthogonal lifted from D6
ρ162200-1-12-12-2-2000-12-1-1-1-111-21111-2-12    orthogonal lifted from D6
ρ172200-1-12-1222000-12-1-1-1-1-1-12-1-1-1-12-12    orthogonal lifted from S3
ρ1822002-1-1-1222000-1-12-1-12-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ192200-1-1-122220002-1-1-12-1-1-1-12-12-1-1-1-1    orthogonal lifted from S3
ρ202200-12-1-1222000-1-1-12-1-122-1-1-1-1-1-12-1    orthogonal lifted from S3
ρ2122002-1-1-1-22-2000-1-12-11-2-1111-2-12-111    orthogonal lifted from D6
ρ222200-1-1-12-2-220002-1-1-1-211-1-12-1-21111    orthogonal lifted from D6
ρ232200-12-1-1-2-22000-1-1-1211-22-1-1-1111-21    orthogonal lifted from D6
ρ2422002-1-1-12-2-2000-1-12-1-121111-21-21-1-1    orthogonal lifted from D6
ρ252-2-222222000000-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ262-22-22222000000-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ274-400-2-24-20000002-422000000000000    symplectic lifted from S3×Q8, Schur index 2
ρ284-400-24-2-2000000222-4000000000000    symplectic lifted from S3×Q8, Schur index 2
ρ294-400-2-2-24000000-4222000000000000    symplectic lifted from S3×Q8, Schur index 2
ρ304-4004-2-2-200000022-42000000000000    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)

1200000
0120000
0012000
0001200
0000111
0000112
,
100000
010000
0012000
0001200
000068
0000107
,
1210000
1200000
0012100
0012000
000010
000001
,
0120000
1120000
001000
000100
000010
000001
,
010000
100000
0012100
000100
0000120
0000012

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

Export

Character table of Q8×C3⋊S3 in TeX

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