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G = C22×C13⋊C6  order 312 = 23·3·13

Direct product of C22 and C13⋊C6

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

Aliases: C22×C13⋊C6, D26⋊3C6, C26⋊(C2×C6), C13⋊C3⋊C23, D13⋊(C2×C6), C13⋊(C22×C6), (C2×C26)⋊4C6, (C22×D13)⋊3C3, (C2×C13⋊C3)⋊C22, (C22×C13⋊C3)⋊2C2, SmallGroup(312,49)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C13 — C22×C13⋊C6
Chief series
C1 — C13 — C13⋊C3 — C13⋊C6 — C2×C13⋊C6 — C22×C13⋊C6
Lower central
C13 — C22×C13⋊C6
Upper central
C1 — C22

Generators and relations for C22×C13⋊C6
 G = < a,b,c,d | a2=b2=c13=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c10 >

Subgroups: 388 in 64 conjugacy classes, 37 normal (8 characteristic)
C1, C2, C2, C3, C22, C22, C6, C23, C2×C6, C13, C22×C6, D13, C26, C13⋊C3, D26, C2×C26, C13⋊C6, C2×C13⋊C3, C22×D13, C2×C13⋊C6, C22×C13⋊C3, C22×C13⋊C6
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C22×C6, C13⋊C6, C2×C13⋊C6, C22×C13⋊C6

Smallest permutation representation of C22×C13⋊C6
►On 52 points
Generators in S52
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B6A···6N13A13B26A···26F
order12222222336···6131326···26
size111113131313131313···13666···6

32 irreducible representations

dim11111166
type+++++
imageC1C2C2C3C6C6C13⋊C6C2×C13⋊C6
kernelC22×C13⋊C6C2×C13⋊C6C22×C13⋊C3C22×D13D26C2×C26C22C2
# reps161212226

Matrix representation of C22×C13⋊C6 ►in GL8(𝔽79)

780000000
078000000
00100000
00010000
00001000
00000100
00000010
00000001
,
780000000
01000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00637764776378
00100000
00010000
00001000
00000100
00000010
,
550000000
055000000
00100000
00614677636062
0016181611717
00000010
00010000
0017171161816

G:=sub<GL(8,GF(79))| [78,0,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[78,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,63,1,0,0,0,0,0,0,77,0,1,0,0,0,0,0,64,0,0,1,0,0,0,0,77,0,0,0,1,0,0,0,63,0,0,0,0,1,0,0,78,0,0,0,0,0],[55,0,0,0,0,0,0,0,0,55,0,0,0,0,0,0,0,0,1,61,16,0,0,17,0,0,0,46,18,0,1,17,0,0,0,77,16,0,0,1,0,0,0,63,1,0,0,16,0,0,0,60,17,1,0,18,0,0,0,62,17,0,0,16] >;
 

C22×C13⋊C6 in GAP, Magma, Sage, TeX 

C_2^2\times C_{13}\rtimes C_6
 
% in TeX
 
G:=Group("C2^2xC13:C6");
 
// GroupNames label
 
G:=SmallGroup(312,49);
 
// by ID
 
G=gap.SmallGroup(312,49);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-3,-13,7204,244]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^13=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^10>;
 
// generators/relations
 

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