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G = C132C24order 312 = 23·3·13

The semidirect product of C13 and C24 acting via C24/C4=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C132C24, C52.2C6, C26.2C12, C132C8⋊C3, C13⋊C32C8, C4.2(C13⋊C6), C2.(C26.C6), (C4×C13⋊C3).2C2, (C2×C13⋊C3).2C4, SmallGroup(312,1)

Series: Derived Chief Lower central Upper central

C1C13 — C132C24
C1C13C26C52C4×C13⋊C3 — C132C24
C13 — C132C24
C1C4

Generators and relations for C132C24
 G = < a,b | a13=b24=1, bab-1=a10 >

13C3
13C6
13C8
13C12
13C24

Smallest permutation representation of C132C24
On 104 points
Generators in S104
(1 99 10 83 70 26 18 37 45 91 62 53 78)(2 63 38 71 100 54 46 27 11 79 92 19 84)(3 93 28 101 64 20 12 55 39 85 80 47 72)(4 57 56 65 94 48 40 21 29 73 86 13 102)(5 87 22 95 58 14 30 49 33 103 74 41 66)(6 75 50 59 88 42 34 15 23 67 104 31 96)(7 81 16 89 76 32 24 43 51 97 68 35 60)(8 69 44 77 82 36 52 9 17 61 98 25 90)
(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)| (1,99,10,83,70,26,18,37,45,91,62,53,78)(2,63,38,71,100,54,46,27,11,79,92,19,84)(3,93,28,101,64,20,12,55,39,85,80,47,72)(4,57,56,65,94,48,40,21,29,73,86,13,102)(5,87,22,95,58,14,30,49,33,103,74,41,66)(6,75,50,59,88,42,34,15,23,67,104,31,96)(7,81,16,89,76,32,24,43,51,97,68,35,60)(8,69,44,77,82,36,52,9,17,61,98,25,90), (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( (1,99,10,83,70,26,18,37,45,91,62,53,78)(2,63,38,71,100,54,46,27,11,79,92,19,84)(3,93,28,101,64,20,12,55,39,85,80,47,72)(4,57,56,65,94,48,40,21,29,73,86,13,102)(5,87,22,95,58,14,30,49,33,103,74,41,66)(6,75,50,59,88,42,34,15,23,67,104,31,96)(7,81,16,89,76,32,24,43,51,97,68,35,60)(8,69,44,77,82,36,52,9,17,61,98,25,90), (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([(1,99,10,83,70,26,18,37,45,91,62,53,78),(2,63,38,71,100,54,46,27,11,79,92,19,84),(3,93,28,101,64,20,12,55,39,85,80,47,72),(4,57,56,65,94,48,40,21,29,73,86,13,102),(5,87,22,95,58,14,30,49,33,103,74,41,66),(6,75,50,59,88,42,34,15,23,67,104,31,96),(7,81,16,89,76,32,24,43,51,97,68,35,60),(8,69,44,77,82,36,52,9,17,61,98,25,90)], [(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)])

32 conjugacy classes

class 1  2 3A3B4A4B6A6B8A8B8C8D12A12B12C12D13A13B24A···24H26A26B52A52B52C52D
order12334466888812121212131324···24262652525252
size11131311131313131313131313136613···13666666

32 irreducible representations

dim11111111666
type+++-
imageC1C2C3C4C6C8C12C24C13⋊C6C26.C6C132C24
kernelC132C24C4×C13⋊C3C132C8C2×C13⋊C3C52C13⋊C3C26C13C4C2C1
# reps11222448224

Matrix representation of C132C24 in GL6(𝔽313)

010000
001000
000100
000010
000001
312110311111311110
,
16372153232306232
213305591203304
6281599197250
153232747984232
811723423981160
91621031219152

G:=sub<GL(6,GF(313))| [0,0,0,0,0,312,1,0,0,0,0,110,0,1,0,0,0,311,0,0,1,0,0,111,0,0,0,1,0,311,0,0,0,0,1,110],[163,213,6,153,81,9,72,305,28,232,17,162,153,59,159,74,234,10,232,1,91,79,239,312,306,203,97,84,81,19,232,304,250,232,160,152] >;

C132C24 in GAP, Magma, Sage, TeX

C_{13}\rtimes_2C_{24}
% in TeX

G:=Group("C13:2C24");
// GroupNames label

G:=SmallGroup(312,1);
// by ID

G=gap.SmallGroup(312,1);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-13,30,42,7204,909]);
// Polycyclic

G:=Group<a,b|a^13=b^24=1,b*a*b^-1=a^10>;
// generators/relations

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Subgroup lattice of C132C24 in TeX

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