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## G = C7×C16⋊C4order 448 = 26·7

### Direct product of C7 and C16⋊C4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C16⋊C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C56 — C7×M5(2) — C7×C16⋊C4
 Lower central C1 — C4 — C7×C16⋊C4
 Upper central C1 — C28 — C7×C16⋊C4

Generators and relations for C7×C16⋊C4
G = < a,b,c | a7=b16=c4=1, ab=ba, ac=ca, cbc-1=b13 >

Smallest permutation representation of C7×C16⋊C4
On 112 points
Generators in S112
(1 72 83 39 23 106 58)(2 73 84 40 24 107 59)(3 74 85 41 25 108 60)(4 75 86 42 26 109 61)(5 76 87 43 27 110 62)(6 77 88 44 28 111 63)(7 78 89 45 29 112 64)(8 79 90 46 30 97 49)(9 80 91 47 31 98 50)(10 65 92 48 32 99 51)(11 66 93 33 17 100 52)(12 67 94 34 18 101 53)(13 68 95 35 19 102 54)(14 69 96 36 20 103 55)(15 70 81 37 21 104 56)(16 71 82 38 22 105 57)
(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)
(2 6 10 14)(3 11)(4 16 12 8)(7 15)(17 25)(18 30 26 22)(20 24 28 32)(21 29)(33 41)(34 46 42 38)(36 40 44 48)(37 45)(49 61 57 53)(51 55 59 63)(52 60)(56 64)(65 69 73 77)(66 74)(67 79 75 71)(70 78)(81 89)(82 94 90 86)(84 88 92 96)(85 93)(97 109 105 101)(99 103 107 111)(100 108)(104 112)

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

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

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

154 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 4E 7A ··· 7F 8A 8B 8C 8D 8E 8F 14A ··· 14F 14G ··· 14L 16A ··· 16H 28A ··· 28L 28M ··· 28R 28S ··· 28AD 56A ··· 56X 56Y ··· 56AJ 112A ··· 112AV order 1 2 2 4 4 4 4 4 7 ··· 7 8 8 8 8 8 8 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 1 1 2 4 4 1 ··· 1 2 2 2 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + image C1 C2 C2 C4 C4 C4 C7 C14 C14 C28 C28 C28 M4(2) M4(2) C7×M4(2) C7×M4(2) C16⋊C4 C7×C16⋊C4 kernel C7×C16⋊C4 C7×C8⋊C4 C7×M5(2) C112 C4×C28 C2×C56 C16⋊C4 C8⋊C4 M5(2) C16 C42 C2×C8 C28 C2×C14 C4 C22 C7 C1 # reps 1 1 2 8 2 2 6 6 12 48 12 12 2 2 12 12 2 12

Matrix representation of C7×C16⋊C4 in GL4(𝔽113) generated by

 30 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 0 0 1 0 112 0 15 2 0 15 0 0 57 57 57 0
,
 1 0 0 0 0 112 0 0 0 0 98 0 106 0 112 15
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,0,57,0,0,15,57,1,15,0,57,0,2,0,0],[1,0,0,106,0,112,0,0,0,0,98,112,0,0,0,15] >;

C7×C16⋊C4 in GAP, Magma, Sage, TeX

C_7\times C_{16}\rtimes C_4
% in TeX

G:=Group("C7xC16:C4");
// GroupNames label

G:=SmallGroup(448,151);
// by ID

G=gap.SmallGroup(448,151);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,1597,400,3538,136,9804,124]);
// Polycyclic

G:=Group<a,b,c|a^7=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations

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