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G = C16×C7⋊C3order 336 = 24·3·7

Direct product of C16 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C16×C7⋊C3, C112⋊C3, C72C48, C56.4C6, C28.5C12, C14.2C24, C2.(C8×C7⋊C3), C4.2(C4×C7⋊C3), C8.2(C2×C7⋊C3), (C2×C7⋊C3).2C8, (C4×C7⋊C3).5C4, (C8×C7⋊C3).4C2, SmallGroup(336,2)

Series: Derived Chief Lower central Upper central

C1C7 — C16×C7⋊C3
C1C7C14C28C56C8×C7⋊C3 — C16×C7⋊C3
C7 — C16×C7⋊C3
C1C16

Generators and relations for C16×C7⋊C3
 G = < a,b,c | a16=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
7C6
7C12
7C24
7C48

Smallest permutation representation of C16×C7⋊C3
On 112 points
Generators in S112
(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)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 74 54 84 22 42 107)(2 75 55 85 23 43 108)(3 76 56 86 24 44 109)(4 77 57 87 25 45 110)(5 78 58 88 26 46 111)(6 79 59 89 27 47 112)(7 80 60 90 28 48 97)(8 65 61 91 29 33 98)(9 66 62 92 30 34 99)(10 67 63 93 31 35 100)(11 68 64 94 32 36 101)(12 69 49 95 17 37 102)(13 70 50 96 18 38 103)(14 71 51 81 19 39 104)(15 72 52 82 20 40 105)(16 73 53 83 21 41 106)
(17 69 49)(18 70 50)(19 71 51)(20 72 52)(21 73 53)(22 74 54)(23 75 55)(24 76 56)(25 77 57)(26 78 58)(27 79 59)(28 80 60)(29 65 61)(30 66 62)(31 67 63)(32 68 64)(33 91 98)(34 92 99)(35 93 100)(36 94 101)(37 95 102)(38 96 103)(39 81 104)(40 82 105)(41 83 106)(42 84 107)(43 85 108)(44 86 109)(45 87 110)(46 88 111)(47 89 112)(48 90 97)

G:=sub<Sym(112)| (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,74,54,84,22,42,107)(2,75,55,85,23,43,108)(3,76,56,86,24,44,109)(4,77,57,87,25,45,110)(5,78,58,88,26,46,111)(6,79,59,89,27,47,112)(7,80,60,90,28,48,97)(8,65,61,91,29,33,98)(9,66,62,92,30,34,99)(10,67,63,93,31,35,100)(11,68,64,94,32,36,101)(12,69,49,95,17,37,102)(13,70,50,96,18,38,103)(14,71,51,81,19,39,104)(15,72,52,82,20,40,105)(16,73,53,83,21,41,106), (17,69,49)(18,70,50)(19,71,51)(20,72,52)(21,73,53)(22,74,54)(23,75,55)(24,76,56)(25,77,57)(26,78,58)(27,79,59)(28,80,60)(29,65,61)(30,66,62)(31,67,63)(32,68,64)(33,91,98)(34,92,99)(35,93,100)(36,94,101)(37,95,102)(38,96,103)(39,81,104)(40,82,105)(41,83,106)(42,84,107)(43,85,108)(44,86,109)(45,87,110)(46,88,111)(47,89,112)(48,90,97)>;

G:=Group( (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,74,54,84,22,42,107)(2,75,55,85,23,43,108)(3,76,56,86,24,44,109)(4,77,57,87,25,45,110)(5,78,58,88,26,46,111)(6,79,59,89,27,47,112)(7,80,60,90,28,48,97)(8,65,61,91,29,33,98)(9,66,62,92,30,34,99)(10,67,63,93,31,35,100)(11,68,64,94,32,36,101)(12,69,49,95,17,37,102)(13,70,50,96,18,38,103)(14,71,51,81,19,39,104)(15,72,52,82,20,40,105)(16,73,53,83,21,41,106), (17,69,49)(18,70,50)(19,71,51)(20,72,52)(21,73,53)(22,74,54)(23,75,55)(24,76,56)(25,77,57)(26,78,58)(27,79,59)(28,80,60)(29,65,61)(30,66,62)(31,67,63)(32,68,64)(33,91,98)(34,92,99)(35,93,100)(36,94,101)(37,95,102)(38,96,103)(39,81,104)(40,82,105)(41,83,106)(42,84,107)(43,85,108)(44,86,109)(45,87,110)(46,88,111)(47,89,112)(48,90,97) );

G=PermutationGroup([[(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,74,54,84,22,42,107),(2,75,55,85,23,43,108),(3,76,56,86,24,44,109),(4,77,57,87,25,45,110),(5,78,58,88,26,46,111),(6,79,59,89,27,47,112),(7,80,60,90,28,48,97),(8,65,61,91,29,33,98),(9,66,62,92,30,34,99),(10,67,63,93,31,35,100),(11,68,64,94,32,36,101),(12,69,49,95,17,37,102),(13,70,50,96,18,38,103),(14,71,51,81,19,39,104),(15,72,52,82,20,40,105),(16,73,53,83,21,41,106)], [(17,69,49),(18,70,50),(19,71,51),(20,72,52),(21,73,53),(22,74,54),(23,75,55),(24,76,56),(25,77,57),(26,78,58),(27,79,59),(28,80,60),(29,65,61),(30,66,62),(31,67,63),(32,68,64),(33,91,98),(34,92,99),(35,93,100),(36,94,101),(37,95,102),(38,96,103),(39,81,104),(40,82,105),(41,83,106),(42,84,107),(43,85,108),(44,86,109),(45,87,110),(46,88,111),(47,89,112),(48,90,97)]])

80 conjugacy classes

class 1  2 3A3B4A4B6A6B7A7B8A8B8C8D12A12B12C12D14A14B16A···16H24A···24H28A28B28C28D48A···48P56A···56H112A···112P
order1233446677888812121212141416···1624···242828282848···4856···56112···112
size117711773311117777331···17···733337···73···33···3

80 irreducible representations

dim111111111133333
type++
imageC1C2C3C4C6C8C12C16C24C48C7⋊C3C2×C7⋊C3C4×C7⋊C3C8×C7⋊C3C16×C7⋊C3
kernelC16×C7⋊C3C8×C7⋊C3C112C4×C7⋊C3C56C2×C7⋊C3C28C7⋊C3C14C7C16C8C4C2C1
# reps11222448816224816

Matrix representation of C16×C7⋊C3 in GL4(𝔽337) generated by

40000
025200
002520
000252
,
1000
01241251
0100
0010
,
128000
0100
0212336336
0010
G:=sub<GL(4,GF(337))| [40,0,0,0,0,252,0,0,0,0,252,0,0,0,0,252],[1,0,0,0,0,124,1,0,0,125,0,1,0,1,0,0],[128,0,0,0,0,1,212,0,0,0,336,1,0,0,336,0] >;

C16×C7⋊C3 in GAP, Magma, Sage, TeX

C_{16}\times C_7\rtimes C_3
% in TeX

G:=Group("C16xC7:C3");
// GroupNames label

G:=SmallGroup(336,2);
// by ID

G=gap.SmallGroup(336,2);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-2,-7,36,50,69,1739]);
// Polycyclic

G:=Group<a,b,c|a^16=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C16×C7⋊C3 in TeX

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