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G = C13⋊C24  order 312 = 23·3·13

The semidirect product of C13 and C24 acting via C24/C2=C12

metacyclic, supersoluble, monomial, Z-group

Aliases: C13⋊C24, C2.F13, C26.C12, Dic13.1C6, C13⋊C3⋊C8, C13⋊C8⋊C3, C26.C6.1C2, (C2×C13⋊C3).C4, SmallGroup(312,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C13 — C13⋊C24
Chief series
C1 — C13 — C26 — Dic13 — C26.C6 — C13⋊C24
Lower central
C13 — C13⋊C24
Upper central
C1 — C2

Generators and relations for C13⋊C24
 G = < a,b | a13=b24=1, bab-1=a2 >

C1
C2
13C3
C13
13C4
13C6
C26
C13⋊C3
13C8
13C12
Dic13
C2×C13⋊C3
13C24
C13⋊C8
C26.C6
C13⋊C24

Character table of C13⋊C24

 class 123A3B4A4B6A6B8A8B8C8D12A12B12C12D1324A24B24C24D24E24F24G24H26
 size 11131313131313131313131313131312131313131313131312
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1-111111-1-1-1-1-1-1-1-11    linear of order 2
ρ311ζ3ζ3211ζ3ζ321111ζ32ζ32ζ3ζ31ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ31    linear of order 3
ρ411ζ32ζ311ζ32ζ31111ζ3ζ3ζ32ζ321ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ321    linear of order 3
ρ511ζ32ζ311ζ32ζ3-1-1-1-1ζ3ζ3ζ32ζ321ζ65ζ6ζ6ζ6ζ65ζ65ζ65ζ61    linear of order 6
ρ611ζ3ζ3211ζ3ζ32-1-1-1-1ζ32ζ32ζ3ζ31ζ6ζ65ζ65ζ65ζ6ζ6ζ6ζ651    linear of order 6
ρ71111-1-111-iii-i-1-1-1-11-ii-i-iii-ii1    linear of order 4
ρ81111-1-111i-i-ii-1-1-1-11i-iii-i-ii-i1    linear of order 4
ρ91-111-ii-1-1ζ8ζ87ζ83ζ85i-i-ii1ζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87-1    linear of order 8
ρ101-111i-i-1-1ζ83ζ85ζ8ζ87-iii-i1ζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85-1    linear of order 8
ρ111-111i-i-1-1ζ87ζ8ζ85ζ83-iii-i1ζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8-1    linear of order 8
ρ121-111-ii-1-1ζ85ζ83ζ87ζ8i-i-ii1ζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83-1    linear of order 8
ρ1311ζ3ζ32-1-1ζ3ζ32i-i-iiζ6ζ6ζ65ζ651ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ31    linear of order 12
ρ1411ζ3ζ32-1-1ζ3ζ32-iii-iζ6ζ6ζ65ζ651ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ31    linear of order 12
ρ1511ζ32ζ3-1-1ζ32ζ3i-i-iiζ65ζ65ζ6ζ61ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ321    linear of order 12
ρ1611ζ32ζ3-1-1ζ32ζ3-iii-iζ65ζ65ζ6ζ61ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ321    linear of order 12
ρ171-1ζ3ζ32-iiζ65ζ6ζ85ζ83ζ87ζ8ζ82ζ32ζ86ζ32ζ86ζ3ζ82ζ31ζ85ζ32ζ87ζ3ζ8ζ3ζ85ζ3ζ83ζ32ζ87ζ32ζ8ζ32ζ83ζ3-1    linear of order 24
ρ181-1ζ32ζ3-iiζ6ζ65ζ85ζ83ζ87ζ8ζ82ζ3ζ86ζ3ζ86ζ32ζ82ζ321ζ85ζ3ζ87ζ32ζ8ζ32ζ85ζ32ζ83ζ3ζ87ζ3ζ8ζ3ζ83ζ32-1    linear of order 24
ρ191-1ζ3ζ32-iiζ65ζ6ζ8ζ87ζ83ζ85ζ82ζ32ζ86ζ32ζ86ζ3ζ82ζ31ζ8ζ32ζ83ζ3ζ85ζ3ζ8ζ3ζ87ζ32ζ83ζ32ζ85ζ32ζ87ζ3-1    linear of order 24
ρ201-1ζ3ζ32i-iζ65ζ6ζ83ζ85ζ8ζ87ζ86ζ32ζ82ζ32ζ82ζ3ζ86ζ31ζ83ζ32ζ8ζ3ζ87ζ3ζ83ζ3ζ85ζ32ζ8ζ32ζ87ζ32ζ85ζ3-1    linear of order 24
ρ211-1ζ3ζ32i-iζ65ζ6ζ87ζ8ζ85ζ83ζ86ζ32ζ82ζ32ζ82ζ3ζ86ζ31ζ87ζ32ζ85ζ3ζ83ζ3ζ87ζ3ζ8ζ32ζ85ζ32ζ83ζ32ζ8ζ3-1    linear of order 24
ρ221-1ζ32ζ3i-iζ6ζ65ζ83ζ85ζ8ζ87ζ86ζ3ζ82ζ3ζ82ζ32ζ86ζ321ζ83ζ3ζ8ζ32ζ87ζ32ζ83ζ32ζ85ζ3ζ8ζ3ζ87ζ3ζ85ζ32-1    linear of order 24
ρ231-1ζ32ζ3i-iζ6ζ65ζ87ζ8ζ85ζ83ζ86ζ3ζ82ζ3ζ82ζ32ζ86ζ321ζ87ζ3ζ85ζ32ζ83ζ32ζ87ζ32ζ8ζ3ζ85ζ3ζ83ζ3ζ8ζ32-1    linear of order 24
ρ241-1ζ32ζ3-iiζ6ζ65ζ8ζ87ζ83ζ85ζ82ζ3ζ86ζ3ζ86ζ32ζ82ζ321ζ8ζ3ζ83ζ32ζ85ζ32ζ8ζ32ζ87ζ3ζ83ζ3ζ85ζ3ζ87ζ32-1    linear of order 24
ρ25121200000000000000-100000000-1    orthogonal lifted from F13
ρ2612-1200000000000000-1000000001    symplectic faithful, Schur index 2

Smallest permutation representation of C13⋊C24
►On 104 points
Generators in S104
(1 87 12 95 73 28 20 39 47 103 65 55 57)(2 13 74 21 48 66 58 88 96 29 40 104 56)(3 75 49 59 97 41 33 14 22 67 89 30 81)(4 50 98 34 23 90 82 76 60 42 15 68 31)(5 99 24 83 61 16 32 51 35 91 77 43 69)(6 25 62 9 36 78 70 100 84 17 52 92 44)(7 63 37 71 85 53 45 26 10 79 101 18 93)(8 38 86 46 11 102 94 64 72 54 27 80 19)

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

Matrix representation of C13⋊C24 ►in GL12(𝔽313)

010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
312312312312312312312312312312312312
,
22520139200139139243243200243
24302252013920013913924324320
90700702959020990702092090
1041041941741041748619401941740
13913924324320024302252013920
1941740010410419417410417486194
108227108882272271818108881888
70295902099070209209009070
29322329320501190293119119223223
09070070295902099070209209
0104104194174104174861940194174
29311911922322302932232932050119

G:=sub<GL(12,GF(313))| [0,0,0,0,0,0,0,0,0,0,0,312,1,0,0,0,0,0,0,0,0,0,0,312,0,1,0,0,0,0,0,0,0,0,0,312,0,0,1,0,0,0,0,0,0,0,0,312,0,0,0,1,0,0,0,0,0,0,0,312,0,0,0,0,1,0,0,0,0,0,0,312,0,0,0,0,0,1,0,0,0,0,0,312,0,0,0,0,0,0,1,0,0,0,0,312,0,0,0,0,0,0,0,1,0,0,0,312,0,0,0,0,0,0,0,0,1,0,0,312,0,0,0,0,0,0,0,0,0,1,0,312,0,0,0,0,0,0,0,0,0,0,1,312],[225,243,90,104,139,194,108,70,293,0,0,293,20,0,70,104,139,174,227,295,223,90,104,119,139,225,0,194,243,0,108,90,293,70,104,119,20,20,70,174,243,0,88,209,205,0,194,223,0,139,295,104,20,104,227,90,0,70,174,223,139,20,90,174,0,104,227,70,119,295,104,0,139,0,209,86,243,194,18,209,0,90,174,293,243,139,90,194,0,174,18,209,293,209,86,223,243,139,70,0,225,104,108,0,119,90,194,293,20,243,209,194,20,174,88,0,119,70,0,205,0,243,209,174,139,86,18,90,223,209,194,0,243,20,0,0,20,194,88,70,223,209,174,119] >;
 

C13⋊C24 in GAP, Magma, Sage, TeX 

C_{13}\rtimes C_{24}
 
% in TeX
 
G:=Group("C13:C24");
 
// GroupNames label
 
G:=SmallGroup(312,7);
 
// by ID
 
G=gap.SmallGroup(312,7);
 
# by ID
 
G:=PCGroup([5,-2,-3,-2,-2,-13,30,42,4804,909,1214]);
 
// Polycyclic
 
G:=Group<a,b|a^13=b^24=1,b*a*b^-1=a^2>;
 
// generators/relations
 

Export

Subgroup lattice of C13⋊C24 in TeX
Character table of C13⋊C24 in TeX

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