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G = C7⋊C48order 336 = 24·3·7

The semidirect product of C7 and C48 acting via C48/C8=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C7⋊C48, C14.C24, C8.2F7, C56.2C6, C28.2C12, C7⋊C3⋊C16, C7⋊C16⋊C3, C2.(C7⋊C24), C4.2(C7⋊C12), (C2×C7⋊C3).C8, (C4×C7⋊C3).2C4, (C8×C7⋊C3).2C2, SmallGroup(336,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C48
C1C7C14C28C56C8×C7⋊C3 — C7⋊C48
C7 — C7⋊C48
C1C8

Generators and relations for C7⋊C48
 G = < a,b | a7=b48=1, bab-1=a5 >

7C3
7C6
7C12
7C16
7C24
7C48

Smallest permutation representation of C7⋊C48
On 112 points
Generators in S112
(1 29 61 86 45 102 70)(2 103 87 30 71 46 62)(3 47 31 104 63 72 88)(4 73 105 48 89 64 32)(5 17 49 74 33 90 106)(6 91 75 18 107 34 50)(7 35 19 92 51 108 76)(8 109 93 36 77 52 20)(9 53 37 110 21 78 94)(10 79 111 54 95 22 38)(11 23 55 80 39 96 112)(12 97 81 24 65 40 56)(13 41 25 98 57 66 82)(14 67 99 42 83 58 26)(15 59 43 68 27 84 100)(16 85 69 60 101 28 44)
(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)

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

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

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

56 conjugacy classes

class 1  2 3A3B4A4B6A6B 7 8A8B8C8D12A12B12C12D 14 16A···16H24A···24H28A28B48A···48P56A56B56C56D
order1233446678888121212121416···1624···24282848···4856565656
size1177117761111777767···77···7667···76666

56 irreducible representations

dim11111111116666
type+++-
imageC1C2C3C4C6C8C12C16C24C48F7C7⋊C12C7⋊C24C7⋊C48
kernelC7⋊C48C8×C7⋊C3C7⋊C16C4×C7⋊C3C56C2×C7⋊C3C28C7⋊C3C14C7C8C4C2C1
# reps112224488161124

Matrix representation of C7⋊C48 in GL6(𝔽337)

010000
001000
000100
000010
000001
336336336336336336
,
22702272272340
22722723400227
711011001100
02270227227234
110007110110
071101100110

G:=sub<GL(6,GF(337))| [0,0,0,0,0,336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336],[227,227,7,0,110,0,0,227,110,227,0,7,227,234,110,0,0,110,227,0,0,227,7,110,234,0,110,227,110,0,0,227,0,234,110,110] >;

C7⋊C48 in GAP, Magma, Sage, TeX

C_7\rtimes C_{48}
% in TeX

G:=Group("C7:C48");
// GroupNames label

G:=SmallGroup(336,1);
// by ID

G=gap.SmallGroup(336,1);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-2,-7,36,50,69,10373,3467]);
// Polycyclic

G:=Group<a,b|a^7=b^48=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C7⋊C48 in TeX

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