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G = C4×C13⋊C6order 312 = 23·3·13

Direct product of C4 and C13⋊C6

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C4×C13⋊C6, C522C6, D132C12, D26.2C6, Dic132C6, (C4×D13)⋊C3, C132(C2×C12), C26.C62C2, C26.2(C2×C6), C13⋊C32(C2×C4), (C4×C13⋊C3)⋊2C2, C2.1(C2×C13⋊C6), (C2×C13⋊C6).2C2, (C2×C13⋊C3).2C22, SmallGroup(312,9)

Series: Derived Chief Lower central Upper central

C1C13 — C4×C13⋊C6
C1C13C26C2×C13⋊C3C2×C13⋊C6 — C4×C13⋊C6
C13 — C4×C13⋊C6
C1C4

Generators and relations for C4×C13⋊C6
 G = < a,b,c | a4=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >

13C2
13C2
13C3
13C4
13C22
13C6
13C6
13C6
13C2×C4
13C12
13C2×C6
13C12
13C2×C12

Smallest permutation representation of C4×C13⋊C6
On 52 points
Generators in S52
(1 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)
(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)
(1 14)(2 18 4 26 10 24)(3 22 7 25 6 21)(5 17 13 23 11 15)(8 16 9 20 12 19)(27 40)(28 44 30 52 36 50)(29 48 33 51 32 47)(31 43 39 49 37 41)(34 42 35 46 38 45)

G:=sub<Sym(52)| (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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), (1,14)(2,18,4,26,10,24)(3,22,7,25,6,21)(5,17,13,23,11,15)(8,16,9,20,12,19)(27,40)(28,44,30,52,36,50)(29,48,33,51,32,47)(31,43,39,49,37,41)(34,42,35,46,38,45)>;

G:=Group( (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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), (1,14)(2,18,4,26,10,24)(3,22,7,25,6,21)(5,17,13,23,11,15)(8,16,9,20,12,19)(27,40)(28,44,30,52,36,50)(29,48,33,51,32,47)(31,43,39,49,37,41)(34,42,35,46,38,45) );

G=PermutationGroup([(1,40,14,27),(2,41,15,28),(3,42,16,29),(4,43,17,30),(5,44,18,31),(6,45,19,32),(7,46,20,33),(8,47,21,34),(9,48,22,35),(10,49,23,36),(11,50,24,37),(12,51,25,38),(13,52,26,39)], [(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)], [(1,14),(2,18,4,26,10,24),(3,22,7,25,6,21),(5,17,13,23,11,15),(8,16,9,20,12,19),(27,40),(28,44,30,52,36,50),(29,48,33,51,32,47),(31,43,39,49,37,41),(34,42,35,46,38,45)])

32 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F12A···12H13A13B26A26B52A52B52C52D
order12223344446···612···121313262652525252
size111313131311131313···1313···1366666666

32 irreducible representations

dim1111111111666
type++++++
imageC1C2C2C2C3C4C6C6C6C12C13⋊C6C2×C13⋊C6C4×C13⋊C6
kernelC4×C13⋊C6C26.C6C4×C13⋊C3C2×C13⋊C6C4×D13C13⋊C6Dic13C52D26D13C4C2C1
# reps1111242228224

Matrix representation of C4×C13⋊C6 in GL6(𝔽157)

12900000
01290000
00129000
00012900
00001290
00000129
,
9165926690156
9265926690156
9166926690156
9165936690156
9165926790156
9165926691156
,
1155916790155
906692659167
00015600
15600000
9113325649267
92642513391155

G:=sub<GL(6,GF(157))| [129,0,0,0,0,0,0,129,0,0,0,0,0,0,129,0,0,0,0,0,0,129,0,0,0,0,0,0,129,0,0,0,0,0,0,129],[91,92,91,91,91,91,65,65,66,65,65,65,92,92,92,93,92,92,66,66,66,66,67,66,90,90,90,90,90,91,156,156,156,156,156,156],[1,90,0,156,91,92,155,66,0,0,133,64,91,92,0,0,25,25,67,65,156,0,64,133,90,91,0,0,92,91,155,67,0,0,67,155] >;

C4×C13⋊C6 in GAP, Magma, Sage, TeX

C_4\times C_{13}\rtimes C_6
% in TeX

G:=Group("C4xC13:C6");
// GroupNames label

G:=SmallGroup(312,9);
// by ID

G=gap.SmallGroup(312,9);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,66,7204,464]);
// Polycyclic

G:=Group<a,b,c|a^4=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

Export

Subgroup lattice of C4×C13⋊C6 in TeX

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