Copied to
clipboard

?

G = C3×C22.29C24  order 192 = 26·3

Direct product of C3 and C22.29C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C22.29C24, C6.1522+ (1+4), (C2×C12)⋊26D4, C4⋊D4⋊6C6, C4⋊1D4⋊7C6, C42⋊9(C2×C6), C4.16(C6×D4), C22≀C2⋊5C6, C4.4D4⋊6C6, (C4×C12)⋊40C22, C12.323(C2×D4), (C6×D4)⋊36C22, (C22×D4)⋊10C6, C24.17(C2×C6), (C6×Q8)⋊51C22, C22.21(C6×D4), C42⋊C2⋊10C6, (C2×C6).355C24, C6.190(C22×D4), C23.9(C22×C6), (C2×C12).664C23, (C22×C12)⋊48C22, (C22×C6).91C23, (C23×C6).16C22, C22.29(C23×C6), C2.4(C3×2+ (1+4)), (D4×C2×C6)⋊22C2, (C2×C4)⋊4(C3×D4), C4⋊C4⋊14(C2×C6), C2.14(D4×C2×C6), (C2×C4○D4)⋊8C6, (C2×D4)⋊4(C2×C6), (C6×C4○D4)⋊20C2, C22⋊C4⋊4(C2×C6), (C22×C4)⋊9(C2×C6), (C2×Q8)⋊13(C2×C6), (C3×C4⋊D4)⋊33C2, (C3×C4⋊1D4)⋊16C2, (C3×C4⋊C4)⋊70C22, (C2×C6).417(C2×D4), (C3×C22≀C2)⋊13C2, (C3×C4.4D4)⋊26C2, (C2×C4).22(C22×C6), (C3×C42⋊C2)⋊31C2, (C3×C22⋊C4)⋊39C22, SmallGroup(192,1424)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C22.29C24
Chief series
C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊1D4 — C3×C22.29C24
Lower central
C1 — C22 — C3×C22.29C24
Upper central
C1 — C2×C6 — C3×C22.29C24

Subgroups: 610 in 334 conjugacy classes, 162 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C6, C6 [×2], C6 [×8], C2×C4 [×2], C2×C4 [×10], C2×C4 [×4], D4 [×22], Q8 [×2], C23, C23 [×6], C23 [×8], C12 [×4], C12 [×6], C2×C6, C2×C6 [×2], C2×C6 [×28], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×14], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24 [×2], C2×C12 [×2], C2×C12 [×10], C2×C12 [×4], C3×D4 [×22], C3×Q8 [×2], C22×C6, C22×C6 [×6], C22×C6 [×8], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, C4×C12 [×2], C3×C22⋊C4 [×10], C3×C4⋊C4 [×2], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×14], C6×D4 [×4], C6×Q8, C3×C4○D4 [×4], C23×C6 [×2], C22.29C24, C3×C42⋊C2, C3×C22≀C2 [×4], C3×C4⋊D4 [×4], C3×C4.4D4 [×2], C3×C4⋊1D4 [×2], D4×C2×C6, C6×C4○D4, C3×C22.29C24

Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C22×D4, 2+ (1+4) [×2], C6×D4 [×6], C23×C6, C22.29C24, D4×C2×C6, C3×2+ (1+4) [×2], C3×C22.29C24

Generators and relations
 G = < a,b,c,d,e,f,g | a3=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Smallest permutation representation
►On 48 points
Generators in S48
(1 5 19)(2 6 20)(3 7 17)(4 8 18)(9 27 21)(10 28 22)(11 25 23)(12 26 24)(13 46 36)(14 47 33)(15 48 34)(16 45 35)(29 40 43)(30 37 44)(31 38 41)(32 39 42)

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

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

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

(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)

(2 4)(6 8)(10 12)(13 39)(14 38)(15 37)(16 40)(18 20)(22 24)(26 28)(29 35)(30 34)(31 33)(32 36)(41 47)(42 46)(43 45)(44 48)

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
003000
000300
000030
000003
,
100000
010000
0012000
0001200
0000120
0000012
,
1200000
0120000
001000
000100
000010
000001
,
010000
100000
000010
000001
001000
000100
,
100000
010000
000100
0012000
0000012
000010
,
100000
0120000
001000
0001200
000010
0000012
,
100000
010000
001000
000100
0000120
0000012

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
 

66 conjugacy classes

class 1 2A2B2C2D2E2F···2K3A3B4A4B4C4D4E···4J6A···6F6G6H6I6J6K···6V12A···12H12I···12T
order1222222···23344444···46···666666···612···1212···12
size1111224···41122224···41···122224···42···24···4

66 irreducible representations

dim11111111111111112244
type++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6D4C3×D42+ (1+4)C3×2+ (1+4)
kernelC3×C22.29C24C3×C42⋊C2C3×C22≀C2C3×C4⋊D4C3×C4.4D4C3×C4⋊1D4D4×C2×C6C6×C4○D4C22.29C24C42⋊C2C22≀C2C4⋊D4C4.4D4C4⋊1D4C22×D4C2×C4○D4C2×C12C2×C4C6C2
# reps11442211228844224824

In GAP, Magma, Sage, TeX 

C_3\times C_2^2._{29}C_2^4
 
% in TeX
 
G:=Group("C3xC2^2.29C2^4");
 
// GroupNames label
 
G:=SmallGroup(192,1424);
 
// by ID
 
G=gap.SmallGroup(192,1424);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,344,2102,555,1571]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=f^2=g^2=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀