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## G = C16×C7⋊C3order 336 = 24·3·7

### Direct product of C16 and C7⋊C3

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

 Derived series C1 — C7 — C16×C7⋊C3
 Chief series C1 — C7 — C14 — C28 — C56 — C8×C7⋊C3 — C16×C7⋊C3
 Lower central C7 — C16×C7⋊C3
 Upper central C1 — C16

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

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 3A 3B 4A 4B 6A 6B 7A 7B 8A 8B 8C 8D 12A 12B 12C 12D 14A 14B 16A ··· 16H 24A ··· 24H 28A 28B 28C 28D 48A ··· 48P 56A ··· 56H 112A ··· 112P order 1 2 3 3 4 4 6 6 7 7 8 8 8 8 12 12 12 12 14 14 16 ··· 16 24 ··· 24 28 28 28 28 48 ··· 48 56 ··· 56 112 ··· 112 size 1 1 7 7 1 1 7 7 3 3 1 1 1 1 7 7 7 7 3 3 1 ··· 1 7 ··· 7 3 3 3 3 7 ··· 7 3 ··· 3 3 ··· 3

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 type + + image C1 C2 C3 C4 C6 C8 C12 C16 C24 C48 C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 C8×C7⋊C3 C16×C7⋊C3 kernel C16×C7⋊C3 C8×C7⋊C3 C112 C4×C7⋊C3 C56 C2×C7⋊C3 C28 C7⋊C3 C14 C7 C16 C8 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 2 2 4 8 16

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

 40 0 0 0 0 252 0 0 0 0 252 0 0 0 0 252
,
 1 0 0 0 0 124 125 1 0 1 0 0 0 0 1 0
,
 128 0 0 0 0 1 0 0 0 212 336 336 0 0 1 0
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

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