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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

Aliases: C7×C16⋊C4, C162C28, C1124C4, C42.1C28, C28.35C42, M5(2).2C14, C28.33M4(2), (C2×C56).5C4, (C2×C8).2C28, (C4×C28).4C4, C56.87(C2×C4), C4.11(C4×C28), C8.19(C2×C28), C8⋊C4.4C14, C14.9(C8⋊C4), C4.6(C7×M4(2)), (C7×M5(2)).6C2, (C2×C56).308C22, (C2×C14).16M4(2), C22.4(C7×M4(2)), C2.3(C7×C8⋊C4), (C7×C8⋊C4).9C2, (C2×C8).45(C2×C14), (C2×C4).66(C2×C28), (C2×C28).327(C2×C4), SmallGroup(448,151)

Series: Derived Chief Lower central Upper central

C1C4 — C7×C16⋊C4
C1C2C4C2×C4C2×C8C2×C56C7×M5(2) — C7×C16⋊C4
C1C4 — C7×C16⋊C4
C1C28 — 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 >

2C2
4C4
2C14
2C8
2C2×C4
4C28
2C2×C28
2C56

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 2A2B4A4B4C4D4E7A···7F8A8B8C8D8E8F14A···14F14G···14L16A···16H28A···28L28M···28R28S···28AD56A···56X56Y···56AJ112A···112AV
order122444447···788888814···1414···1416···1628···2828···2828···2856···5656···56112···112
size112112441···12222441···12···24···41···12···24···42···24···44···4

154 irreducible representations

dim111111111111222244
type+++
imageC1C2C2C4C4C4C7C14C14C28C28C28M4(2)M4(2)C7×M4(2)C7×M4(2)C16⋊C4C7×C16⋊C4
kernelC7×C16⋊C4C7×C8⋊C4C7×M5(2)C112C4×C28C2×C56C16⋊C4C8⋊C4M5(2)C16C42C2×C8C28C2×C14C4C22C7C1
# reps1128226612481212221212212

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

30000
03000
00300
00030
,
0010
1120152
01500
5757570
,
1000
011200
00980
106011215
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

Export

Subgroup lattice of C7×C16⋊C4 in TeX

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