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G = C2×C3⋊C16order 96 = 25·3

Direct product of C2 and C3⋊C16

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C3⋊C16, C6⋊C16, C12.3C8, C24.3C4, C8.21D6, C8.4Dic3, C24.25C22, C8(C3⋊C16), C4(C3⋊C16), C32(C2×C16), C4.3(C3⋊C8), (C2×C8).9S3, (C2×C6).2C8, C6.8(C2×C8), (C2×C24).12C2, C12.38(C2×C4), (C2×C12).11C4, C22.2(C3⋊C8), (C2×C4).8Dic3, C4.10(C2×Dic3), C2.2(C2×C3⋊C8), SmallGroup(96,18)

Series: Derived Chief Lower central Upper central

C1C3 — C2×C3⋊C16
C1C3C6C12C24C3⋊C16 — C2×C3⋊C16
C3 — C2×C3⋊C16
C1C2×C8

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

3C16
3C16
3C2×C16

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 2A2B2C 3 4A4B4C4D6A6B6C8A···8H12A12B12C12D16A···16P24A···24H
order1222344446668···81212121216···1624···24
size1111211112221···122223···32···2

48 irreducible representations

dim111111112222222
type++++-+-
imageC1C2C2C4C4C8C8C16S3Dic3D6Dic3C3⋊C8C3⋊C8C3⋊C16
kernelC2×C3⋊C16C3⋊C16C2×C24C24C2×C12C12C2×C6C6C2×C8C8C8C2×C4C4C22C2
# reps1212244161111228

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

1000
09600
0010
0001
,
1000
0100
009696
0010
,
18000
09600
001378
006584
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

Export

Subgroup lattice of C2×C3⋊C16 in TeX

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