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## G = C2×C32⋊2C16order 288 = 25·32

### Direct product of C2 and C32⋊2C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32⋊2C16
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2C16 — C2×C32⋊2C16
 Lower central C32 — C2×C32⋊2C16
 Upper central C1 — C2×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 >

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 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A ··· 6F 8A ··· 8H 12A ··· 12H 16A ··· 16P order 1 2 2 2 3 3 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 16 ··· 16 size 1 1 1 1 4 4 1 1 1 1 4 ··· 4 9 ··· 9 4 ··· 4 9 ··· 9

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 C32⋊C4 C32⋊2C8 C2×C32⋊C4 C32⋊2C8 C32⋊2C16 kernel C2×C32⋊2C16 C32⋊2C16 C2×C32⋊4C8 C32⋊4C8 C6×C12 C3×C12 C62 C3×C6 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 2 2 2 2 8

Matrix representation of C2×C322C16 in GL5(𝔽97)

 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 0 0 0 1 0 0 0 0 0 0 1 0 13 29 96 96
,
 1 0 0 0 0 0 0 96 0 0 0 1 96 0 0 0 70 41 0 1 0 83 70 96 96
,
 85 0 0 0 0 0 0 0 96 1 0 13 29 95 96 0 34 44 41 27 0 51 91 41 27

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

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