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G = C2×C3⋊Q16order 96 = 25·3

Direct product of C2 and C3⋊Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C3⋊Q16, C62Q16, C12.20D4, Q8.12D6, C12.16C23, Dic6.10C22, C33(C2×Q16), C6.55(C2×D4), (C2×C4).54D6, (C2×C6).43D4, C3⋊C8.9C22, (C6×Q8).3C2, (C2×Q8).5S3, C4.9(C3⋊D4), C4.16(C22×S3), (C2×Dic6).8C2, (C3×Q8).7C22, (C2×C12).38C22, C22.24(C3⋊D4), (C2×C3⋊C8).6C2, C2.19(C2×C3⋊D4), SmallGroup(96,150)

Series: Derived Chief Lower central Upper central

C1C12 — C2×C3⋊Q16
C1C3C6C12Dic6C2×Dic6 — C2×C3⋊Q16
C3C6C12 — C2×C3⋊Q16
C1C22C2×C4C2×Q8

Generators and relations for C2×C3⋊Q16
 G = < a,b,c,d | a2=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 114 in 60 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], Q8 [×4], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C8, Q16 [×4], C2×Q8, C2×Q8, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C2×Q16, C2×C3⋊C8, C3⋊Q16 [×4], C2×Dic6, C6×Q8, C2×C3⋊Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C3⋊D4 [×2], C22×S3, C2×Q16, C3⋊Q16 [×2], C2×C3⋊D4, C2×C3⋊Q16

Character table of C2×C3⋊Q16

 class 12A2B2C34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D12E12F
 size 11112224412122226666444444
ρ1111111111111111111111111    trivial
ρ21111111-1-111111-1-1-1-11-1-1-1-11    linear of order 2
ρ311-1-111-11-11-1-11-11-1-1111-11-1-1    linear of order 2
ρ411-1-111-1-111-1-11-1-111-11-11-11-1    linear of order 2
ρ5111111111-1-1111-1-1-1-1111111    linear of order 2
ρ61111111-1-1-1-111111111-1-1-1-11    linear of order 2
ρ711-1-111-11-1-11-11-1-111-111-11-1-1    linear of order 2
ρ811-1-111-1-11-11-11-11-1-111-11-11-1    linear of order 2
ρ92222-1222200-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022222-2-200002220000-20000-2    orthogonal lifted from D4
ρ1122-2-2-12-2-22001-110000-11-11-11    orthogonal lifted from D6
ρ1222-2-2-12-22-2001-110000-1-11-111    orthogonal lifted from D6
ρ1322-2-22-220000-22-20000-200002    orthogonal lifted from D4
ρ142222-122-2-200-1-1-10000-11111-1    orthogonal lifted from D6
ρ152-22-22000000-2-222-22-2000000    symplectic lifted from Q16, Schur index 2
ρ162-2-2220000002-2-2-2-222000000    symplectic lifted from Q16, Schur index 2
ρ172-22-22000000-2-22-22-22000000    symplectic lifted from Q16, Schur index 2
ρ182-2-2220000002-2-222-2-2000000    symplectic lifted from Q16, Schur index 2
ρ192222-1-2-20000-1-1-100001-3--3--3-31    complex lifted from C3⋊D4
ρ202222-1-2-20000-1-1-100001--3-3-3--31    complex lifted from C3⋊D4
ρ2122-2-2-1-2200001-1100001--3--3-3-3-1    complex lifted from C3⋊D4
ρ2222-2-2-1-2200001-1100001-3-3--3--3-1    complex lifted from C3⋊D4
ρ234-4-44-2000000-2220000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ244-44-4-200000022-20000000000    symplectic lifted from C3⋊Q16, Schur index 2

Smallest permutation representation of C2×C3⋊Q16
Regular action on 96 points
Generators in S96
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 86)(10 87)(11 88)(12 81)(13 82)(14 83)(15 84)(16 85)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)(65 94)(66 95)(67 96)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 55 91)(2 92 56)(3 49 93)(4 94 50)(5 51 95)(6 96 52)(7 53 89)(8 90 54)(9 39 76)(10 77 40)(11 33 78)(12 79 34)(13 35 80)(14 73 36)(15 37 74)(16 75 38)(17 84 30)(18 31 85)(19 86 32)(20 25 87)(21 88 26)(22 27 81)(23 82 28)(24 29 83)(41 66 57)(42 58 67)(43 68 59)(44 60 69)(45 70 61)(46 62 71)(47 72 63)(48 64 65)
(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)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
(1 80 5 76)(2 79 6 75)(3 78 7 74)(4 77 8 73)(9 55 13 51)(10 54 14 50)(11 53 15 49)(12 52 16 56)(17 63 21 59)(18 62 22 58)(19 61 23 57)(20 60 24 64)(25 69 29 65)(26 68 30 72)(27 67 31 71)(28 66 32 70)(33 89 37 93)(34 96 38 92)(35 95 39 91)(36 94 40 90)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)

G:=sub<Sym(96)| (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,86)(10,87)(11,88)(12,81)(13,82)(14,83)(15,84)(16,85)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,49)(48,50)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93), (1,55,91)(2,92,56)(3,49,93)(4,94,50)(5,51,95)(6,96,52)(7,53,89)(8,90,54)(9,39,76)(10,77,40)(11,33,78)(12,79,34)(13,35,80)(14,73,36)(15,37,74)(16,75,38)(17,84,30)(18,31,85)(19,86,32)(20,25,87)(21,88,26)(22,27,81)(23,82,28)(24,29,83)(41,66,57)(42,58,67)(43,68,59)(44,60,69)(45,70,61)(46,62,71)(47,72,63)(48,64,65), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,80,5,76)(2,79,6,75)(3,78,7,74)(4,77,8,73)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,63,21,59)(18,62,22,58)(19,61,23,57)(20,60,24,64)(25,69,29,65)(26,68,30,72)(27,67,31,71)(28,66,32,70)(33,89,37,93)(34,96,38,92)(35,95,39,91)(36,94,40,90)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)>;

G:=Group( (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,86)(10,87)(11,88)(12,81)(13,82)(14,83)(15,84)(16,85)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,49)(48,50)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93), (1,55,91)(2,92,56)(3,49,93)(4,94,50)(5,51,95)(6,96,52)(7,53,89)(8,90,54)(9,39,76)(10,77,40)(11,33,78)(12,79,34)(13,35,80)(14,73,36)(15,37,74)(16,75,38)(17,84,30)(18,31,85)(19,86,32)(20,25,87)(21,88,26)(22,27,81)(23,82,28)(24,29,83)(41,66,57)(42,58,67)(43,68,59)(44,60,69)(45,70,61)(46,62,71)(47,72,63)(48,64,65), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,80,5,76)(2,79,6,75)(3,78,7,74)(4,77,8,73)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,63,21,59)(18,62,22,58)(19,61,23,57)(20,60,24,64)(25,69,29,65)(26,68,30,72)(27,67,31,71)(28,66,32,70)(33,89,37,93)(34,96,38,92)(35,95,39,91)(36,94,40,90)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87) );

G=PermutationGroup([(1,61),(2,62),(3,63),(4,64),(5,57),(6,58),(7,59),(8,60),(9,86),(10,87),(11,88),(12,81),(13,82),(14,83),(15,84),(16,85),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,73),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,49),(48,50),(65,94),(66,95),(67,96),(68,89),(69,90),(70,91),(71,92),(72,93)], [(1,55,91),(2,92,56),(3,49,93),(4,94,50),(5,51,95),(6,96,52),(7,53,89),(8,90,54),(9,39,76),(10,77,40),(11,33,78),(12,79,34),(13,35,80),(14,73,36),(15,37,74),(16,75,38),(17,84,30),(18,31,85),(19,86,32),(20,25,87),(21,88,26),(22,27,81),(23,82,28),(24,29,83),(41,66,57),(42,58,67),(43,68,59),(44,60,69),(45,70,61),(46,62,71),(47,72,63),(48,64,65)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,80,5,76),(2,79,6,75),(3,78,7,74),(4,77,8,73),(9,55,13,51),(10,54,14,50),(11,53,15,49),(12,52,16,56),(17,63,21,59),(18,62,22,58),(19,61,23,57),(20,60,24,64),(25,69,29,65),(26,68,30,72),(27,67,31,71),(28,66,32,70),(33,89,37,93),(34,96,38,92),(35,95,39,91),(36,94,40,90),(41,86,45,82),(42,85,46,81),(43,84,47,88),(44,83,48,87)])

C2×C3⋊Q16 is a maximal subgroup of
D12.7D4  C3⋊Q16⋊C4  Dic34Q16  Dic3⋊Q16  Dic6.11D4  Q8.11D12  D6⋊Q16  D61Q16  C3⋊C8.D4  Q8.6D12  C42.59D6  C127Q16  D12.37D4  Dic6.37D4  C3⋊C8.29D4  C3⋊C8.6D4  C42.61D6  C42.214D6  C42.65D6  C42.80D6  C12⋊Q16  C123Q16  (C3×Q8).D4  C24.31D4  C24.43D4  Dic6.16D4  Dic33Q16  C24.26D4  D65Q16  C24.37D4  M4(2).16D6  (C2×C6)⋊8Q16  (C3×D4).32D4  C2×S3×Q16  SD16.D6  D12.35C23
C2×C3⋊Q16 is a maximal quotient of
C4⋊C4.230D6  Q85Dic6  C127Q16  (C2×C6).Q16  Dic6.37D4  C3⋊C8.29D4  C12.17D8  C12.9Q16  C12⋊Q16  Dic65Q8  C123Q16  C12.Q16  (C2×C6)⋊8Q16

Matrix representation of C2×C3⋊Q16 in GL4(𝔽73) generated by

72000
07200
00720
00072
,
1000
0100
00072
00172
,
165700
161600
002639
006547
,
127200
726100
004360
001330
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[16,16,0,0,57,16,0,0,0,0,26,65,0,0,39,47],[12,72,0,0,72,61,0,0,0,0,43,13,0,0,60,30] >;

C2×C3⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes Q_{16}
% in TeX

G:=Group("C2xC3:Q16");
// GroupNames label

G:=SmallGroup(96,150);
// by ID

G=gap.SmallGroup(96,150);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,86,579,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of C2×C3⋊Q16 in TeX

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