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G = C113⋊C4order 452 = 22·113

The semidirect product of C113 and C4 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C113⋊C4, D113.C2, SmallGroup(452,3)

Series: Derived Chief Lower central Upper central

C1C113 — C113⋊C4
C1C113D113 — C113⋊C4
C113 — C113⋊C4
C1

Generators and relations for C113⋊C4
 G = < a,b | a113=b4=1, bab-1=a15 >

113C2
113C4

Smallest permutation representation of C113⋊C4
On 113 points: primitive
Generators in S113
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113)
(2 99 113 16)(3 84 112 31)(4 69 111 46)(5 54 110 61)(6 39 109 76)(7 24 108 91)(8 9 107 106)(10 92 105 23)(11 77 104 38)(12 62 103 53)(13 47 102 68)(14 32 101 83)(15 17 100 98)(18 85 97 30)(19 70 96 45)(20 55 95 60)(21 40 94 75)(22 25 93 90)(26 78 89 37)(27 63 88 52)(28 48 87 67)(29 33 86 82)(34 71 81 44)(35 56 80 59)(36 41 79 74)(42 64 73 51)(43 49 72 66)(50 57 65 58)

G:=sub<Sym(113)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113), (2,99,113,16)(3,84,112,31)(4,69,111,46)(5,54,110,61)(6,39,109,76)(7,24,108,91)(8,9,107,106)(10,92,105,23)(11,77,104,38)(12,62,103,53)(13,47,102,68)(14,32,101,83)(15,17,100,98)(18,85,97,30)(19,70,96,45)(20,55,95,60)(21,40,94,75)(22,25,93,90)(26,78,89,37)(27,63,88,52)(28,48,87,67)(29,33,86,82)(34,71,81,44)(35,56,80,59)(36,41,79,74)(42,64,73,51)(43,49,72,66)(50,57,65,58)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113), (2,99,113,16)(3,84,112,31)(4,69,111,46)(5,54,110,61)(6,39,109,76)(7,24,108,91)(8,9,107,106)(10,92,105,23)(11,77,104,38)(12,62,103,53)(13,47,102,68)(14,32,101,83)(15,17,100,98)(18,85,97,30)(19,70,96,45)(20,55,95,60)(21,40,94,75)(22,25,93,90)(26,78,89,37)(27,63,88,52)(28,48,87,67)(29,33,86,82)(34,71,81,44)(35,56,80,59)(36,41,79,74)(42,64,73,51)(43,49,72,66)(50,57,65,58) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113)], [(2,99,113,16),(3,84,112,31),(4,69,111,46),(5,54,110,61),(6,39,109,76),(7,24,108,91),(8,9,107,106),(10,92,105,23),(11,77,104,38),(12,62,103,53),(13,47,102,68),(14,32,101,83),(15,17,100,98),(18,85,97,30),(19,70,96,45),(20,55,95,60),(21,40,94,75),(22,25,93,90),(26,78,89,37),(27,63,88,52),(28,48,87,67),(29,33,86,82),(34,71,81,44),(35,56,80,59),(36,41,79,74),(42,64,73,51),(43,49,72,66),(50,57,65,58)]])

32 conjugacy classes

class 1  2 4A4B113A···113AB
order1244113···113
size11131131134···4

32 irreducible representations

dim1114
type+++
imageC1C2C4C113⋊C4
kernelC113⋊C4D113C113C1
# reps11228

Matrix representation of C113⋊C4 in GL4(𝔽2713) generated by

1866100
2495010
291001
224714761245892
,
28317654491870
167870024071568
2079117551402
236425752232688
G:=sub<GL(4,GF(2713))| [1866,2495,291,2247,1,0,0,1476,0,1,0,1245,0,0,1,892],[283,1678,20,2364,1765,700,791,2575,449,2407,1755,223,1870,1568,1402,2688] >;

C113⋊C4 in GAP, Magma, Sage, TeX

C_{113}\rtimes C_4
% in TeX

G:=Group("C113:C4");
// GroupNames label

G:=SmallGroup(452,3);
// by ID

G=gap.SmallGroup(452,3);
# by ID

G:=PCGroup([3,-2,-2,-113,6,3530,2021]);
// Polycyclic

G:=Group<a,b|a^113=b^4=1,b*a*b^-1=a^15>;
// generators/relations

Export

Subgroup lattice of C113⋊C4 in TeX

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