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## G = C2×C3⋊C16order 96 = 25·3

### Direct product of C2 and C3⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C3⋊C16
 Chief series C1 — C3 — C6 — C12 — C24 — C3⋊C16 — C2×C3⋊C16
 Lower central C3 — C2×C3⋊C16
 Upper central C1 — C2×C8

Generators and relations for C2×C3⋊C16
G = < a,b,c | a2=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C2×C3⋊C16
Regular action on 96 points
Generators in S96
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 81)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 80)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)
(1 42 31)(2 32 43)(3 44 17)(4 18 45)(5 46 19)(6 20 47)(7 48 21)(8 22 33)(9 34 23)(10 24 35)(11 36 25)(12 26 37)(13 38 27)(14 28 39)(15 40 29)(16 30 41)(49 84 75)(50 76 85)(51 86 77)(52 78 87)(53 88 79)(54 80 89)(55 90 65)(56 66 91)(57 92 67)(58 68 93)(59 94 69)(60 70 95)(61 96 71)(62 72 81)(63 82 73)(64 74 83)
(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)

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

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

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

C2×C3⋊C16 is a maximal subgroup of
C24.C8  C12⋊C16  C6.6D16  C6.SD32  C6.D16  C6.Q32  C24.7Q8  Dic12.C4  Dic3×C16  Dic3⋊C16  C4810C4  D6⋊C16  Dic6.C8  C24.98D4  C24.99D4  D81Dic3  C6.5Q32  C24.41D4  S3×C2×C16  C16.12D6  C24.78C23  Q16.D6  C24.F5
C2×C3⋊C16 is a maximal quotient of
C12⋊C16  C3⋊M6(2)  C24.98D4  C24.F5

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 8A ··· 8H 12A 12B 12C 12D 16A ··· 16P 24A ··· 24H order 1 2 2 2 3 4 4 4 4 6 6 6 8 ··· 8 12 12 12 12 16 ··· 16 24 ··· 24 size 1 1 1 1 2 1 1 1 1 2 2 2 1 ··· 1 2 2 2 2 3 ··· 3 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 S3 Dic3 D6 Dic3 C3⋊C8 C3⋊C8 C3⋊C16 kernel C2×C3⋊C16 C3⋊C16 C2×C24 C24 C2×C12 C12 C2×C6 C6 C2×C8 C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 1 1 1 1 2 2 8

Matrix representation of C2×C3⋊C16 in GL4(𝔽97) generated by

 1 0 0 0 0 96 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 96 96 0 0 1 0
,
 18 0 0 0 0 96 0 0 0 0 13 78 0 0 65 84
G:=sub<GL(4,GF(97))| [1,0,0,0,0,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,96,1,0,0,96,0],[18,0,0,0,0,96,0,0,0,0,13,65,0,0,78,84] >;

C2×C3⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC3:C16");
// GroupNames label

G:=SmallGroup(96,18);
// by ID

G=gap.SmallGroup(96,18);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,50,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^2=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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