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## G = C2×C3⋊Q16order 96 = 25·3

### Direct product of C2 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C3⋊Q16
 Chief series C1 — C3 — C6 — C12 — Dic6 — C2×Dic6 — C2×C3⋊Q16
 Lower central C3 — C6 — C12 — C2×C3⋊Q16
 Upper central C1 — C22 — C2×C4 — C2×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 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 4 4 12 12 2 2 2 6 6 6 6 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 -1 2 2 2 2 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 2 -2 -2 0 0 0 0 2 2 2 0 0 0 0 -2 0 0 0 0 -2 orthogonal lifted from D4 ρ11 2 2 -2 -2 -1 2 -2 -2 2 0 0 1 -1 1 0 0 0 0 -1 1 -1 1 -1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 -1 2 -2 2 -2 0 0 1 -1 1 0 0 0 0 -1 -1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 -2 2 -2 2 0 0 0 0 -2 2 -2 0 0 0 0 -2 0 0 0 0 2 orthogonal lifted from D4 ρ14 2 2 2 2 -1 2 2 -2 -2 0 0 -1 -1 -1 0 0 0 0 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ15 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 2 √2 -√2 √2 -√2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 -√2 -√2 √2 √2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ17 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 2 -√2 √2 -√2 √2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ18 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 √2 √2 -√2 -√2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ19 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 √-3 -√-3 -√-3 √-3 1 complex lifted from C3⋊D4 ρ20 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 -√-3 √-3 √-3 -√-3 1 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 -1 -2 2 0 0 0 0 1 -1 1 0 0 0 0 1 -√-3 -√-3 √-3 √-3 -1 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 -1 -2 2 0 0 0 0 1 -1 1 0 0 0 0 1 √-3 √-3 -√-3 -√-3 -1 complex lifted from C3⋊D4 ρ23 4 -4 -4 4 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2 ρ24 4 -4 4 -4 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 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)])

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

 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 72 0 0 1 72
,
 16 57 0 0 16 16 0 0 0 0 26 39 0 0 65 47
,
 12 72 0 0 72 61 0 0 0 0 43 60 0 0 13 30
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

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