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G = C2×C322C16order 288 = 25·32

Direct product of C2 and C322C16

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×C322C16, C62.3C8, (C3×C6)⋊2C16, (C6×C12).2C4, (C3×C12).5C8, C326(C2×C16), C324C8.9C4, C4.3(C322C8), C324C8.34C22, C22.2(C322C8), (C3×C6).22(C2×C8), C4.19(C2×C32⋊C4), (C3×C12).16(C2×C4), (C2×C4).9(C32⋊C4), C2.1(C2×C322C8), (C2×C324C8).19C2, SmallGroup(288,420)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C322C16
C1C32C3×C6C3×C12C324C8C322C16 — C2×C322C16
C32 — C2×C322C16
C1C2×C4

Generators and relations for C2×C322C16
 G = < a,b,c,d | a2=b3=c3=d16=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

2C3
2C3
2C6
2C6
2C6
2C6
2C6
2C6
9C8
9C8
2C12
2C12
2C12
2C2×C6
2C2×C6
2C12
9C2×C8
9C16
9C16
2C2×C12
2C2×C12
6C3⋊C8
6C3⋊C8
6C3⋊C8
6C3⋊C8
9C2×C16
6C2×C3⋊C8
6C2×C3⋊C8

Smallest permutation representation of C2×C322C16
On 96 points
Generators in S96
(1 64)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 33)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(73 96)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)
(2 25 83)(4 85 27)(6 29 87)(8 89 31)(10 17 91)(12 93 19)(14 21 95)(16 81 23)(34 68 57)(36 59 70)(38 72 61)(40 63 74)(42 76 49)(44 51 78)(46 80 53)(48 55 66)
(1 24 82)(2 25 83)(3 84 26)(4 85 27)(5 28 86)(6 29 87)(7 88 30)(8 89 31)(9 32 90)(10 17 91)(11 92 18)(12 93 19)(13 20 94)(14 21 95)(15 96 22)(16 81 23)(33 67 56)(34 68 57)(35 58 69)(36 59 70)(37 71 60)(38 72 61)(39 62 73)(40 63 74)(41 75 64)(42 76 49)(43 50 77)(44 51 78)(45 79 52)(46 80 53)(47 54 65)(48 55 66)
(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)| (1,64)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (2,25,83)(4,85,27)(6,29,87)(8,89,31)(10,17,91)(12,93,19)(14,21,95)(16,81,23)(34,68,57)(36,59,70)(38,72,61)(40,63,74)(42,76,49)(44,51,78)(46,80,53)(48,55,66), (1,24,82)(2,25,83)(3,84,26)(4,85,27)(5,28,86)(6,29,87)(7,88,30)(8,89,31)(9,32,90)(10,17,91)(11,92,18)(12,93,19)(13,20,94)(14,21,95)(15,96,22)(16,81,23)(33,67,56)(34,68,57)(35,58,69)(36,59,70)(37,71,60)(38,72,61)(39,62,73)(40,63,74)(41,75,64)(42,76,49)(43,50,77)(44,51,78)(45,79,52)(46,80,53)(47,54,65)(48,55,66), (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( (1,64)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (2,25,83)(4,85,27)(6,29,87)(8,89,31)(10,17,91)(12,93,19)(14,21,95)(16,81,23)(34,68,57)(36,59,70)(38,72,61)(40,63,74)(42,76,49)(44,51,78)(46,80,53)(48,55,66), (1,24,82)(2,25,83)(3,84,26)(4,85,27)(5,28,86)(6,29,87)(7,88,30)(8,89,31)(9,32,90)(10,17,91)(11,92,18)(12,93,19)(13,20,94)(14,21,95)(15,96,22)(16,81,23)(33,67,56)(34,68,57)(35,58,69)(36,59,70)(37,71,60)(38,72,61)(39,62,73)(40,63,74)(41,75,64)(42,76,49)(43,50,77)(44,51,78)(45,79,52)(46,80,53)(47,54,65)(48,55,66), (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([(1,64),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,33),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95),(73,96),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87)], [(2,25,83),(4,85,27),(6,29,87),(8,89,31),(10,17,91),(12,93,19),(14,21,95),(16,81,23),(34,68,57),(36,59,70),(38,72,61),(40,63,74),(42,76,49),(44,51,78),(46,80,53),(48,55,66)], [(1,24,82),(2,25,83),(3,84,26),(4,85,27),(5,28,86),(6,29,87),(7,88,30),(8,89,31),(9,32,90),(10,17,91),(11,92,18),(12,93,19),(13,20,94),(14,21,95),(15,96,22),(16,81,23),(33,67,56),(34,68,57),(35,58,69),(36,59,70),(37,71,60),(38,72,61),(39,62,73),(40,63,74),(41,75,64),(42,76,49),(43,50,77),(44,51,78),(45,79,52),(46,80,53),(47,54,65),(48,55,66)], [(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)])

48 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F8A···8H12A···12H16A···16P
order12223344446···68···812···1216···16
size11114411114···49···94···49···9

48 irreducible representations

dim1111111144444
type++++-+-
imageC1C2C2C4C4C8C8C16C32⋊C4C322C8C2×C32⋊C4C322C8C322C16
kernelC2×C322C16C322C16C2×C324C8C324C8C6×C12C3×C12C62C3×C6C2×C4C4C4C22C2
# reps12122441622228

Matrix representation of C2×C322C16 in GL5(𝔽97)

10000
096000
009600
000960
000096
,
10000
01000
00100
00001
013299696
,
10000
009600
019600
0704101
083709696
,
850000
000961
013299596
034444127
051914127

G:=sub<GL(5,GF(97))| [1,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,96],[1,0,0,0,0,0,1,0,0,13,0,0,1,0,29,0,0,0,0,96,0,0,0,1,96],[1,0,0,0,0,0,0,1,70,83,0,96,96,41,70,0,0,0,0,96,0,0,0,1,96],[85,0,0,0,0,0,0,13,34,51,0,0,29,44,91,0,96,95,41,41,0,1,96,27,27] >;

C2×C322C16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2C_{16}
% in TeX

G:=Group("C2xC3^2:2C16");
// GroupNames label

G:=SmallGroup(288,420);
// by ID

G=gap.SmallGroup(288,420);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,58,80,9413,691,12550,2372]);
// Polycyclic

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

Export

Subgroup lattice of C2×C322C16 in TeX

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