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G = C3xC22.29C24order 192 = 26·3

Direct product of C3 and C22.29C24

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

Aliases: C3xC22.29C24, C6.1522+ 1+4, (C2xC12):26D4, C4:D4:6C6, C4:1D4:7C6, C42:9(C2xC6), C4.16(C6xD4), C22wrC2:5C6, C4.4D4:6C6, (C4xC12):40C22, C12.323(C2xD4), (C22xD4):10C6, (C6xD4):36C22, C24.17(C2xC6), (C6xQ8):51C22, C22.21(C6xD4), C42:C2:10C6, (C2xC6).355C24, C6.190(C22xD4), C23.9(C22xC6), (C2xC12).664C23, (C22xC12):48C22, C22.29(C23xC6), (C23xC6).16C22, (C22xC6).91C23, C2.4(C3x2+ 1+4), (D4xC2xC6):22C2, (C2xC4):4(C3xD4), C2.14(D4xC2xC6), C4:C4:14(C2xC6), (C2xC4oD4):8C6, (C2xD4):4(C2xC6), (C6xC4oD4):20C2, C22:C4:4(C2xC6), (C22xC4):9(C2xC6), (C2xQ8):13(C2xC6), (C3xC4:D4):33C2, (C3xC4:1D4):16C2, (C3xC4:C4):70C22, (C2xC6).417(C2xD4), (C3xC22wrC2):13C2, (C3xC4.4D4):26C2, (C2xC4).22(C22xC6), (C3xC42:C2):31C2, (C3xC22:C4):39C22, SmallGroup(192,1424)

Series: Derived Chief Lower central Upper central

C1C22 — C3xC22.29C24
C1C2C22C2xC6C22xC6C6xD4C3xC4:1D4 — C3xC22.29C24
C1C22 — C3xC22.29C24
C1C2xC6 — C3xC22.29C24

Generators and relations for C3xC22.29C24
 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 >

Subgroups: 610 in 334 conjugacy classes, 162 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C2xC12, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C22xC6, C22xC6, C42:C2, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xD4, C2xC4oD4, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C22xC12, C6xD4, C6xD4, C6xD4, C6xQ8, C3xC4oD4, C23xC6, C22.29C24, C3xC42:C2, C3xC22wrC2, C3xC4:D4, C3xC4.4D4, C3xC4:1D4, D4xC2xC6, C6xC4oD4, C3xC22.29C24
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C24, C3xD4, C22xC6, C22xD4, 2+ 1+4, C6xD4, C23xC6, C22.29C24, D4xC2xC6, C3x2+ 1+4, C3xC22.29C24

Smallest permutation representation of C3xC22.29C24
On 48 points
Generators in S48
(1 7 19)(2 8 20)(3 5 17)(4 6 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 42)(30 37 43)(31 38 44)(32 39 41)
(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 27)(6 28)(7 25)(8 26)(13 31)(14 32)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 43)(2 42)(3 41)(4 44)(5 32)(6 31)(7 30)(8 29)(9 33)(10 36)(11 35)(12 34)(13 28)(14 27)(15 26)(16 25)(17 39)(18 38)(19 37)(20 40)(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 29)(14 32)(15 31)(16 30)(18 20)(22 24)(26 28)(33 41)(34 44)(35 43)(36 42)(37 45)(38 48)(39 47)(40 46)
(1 11)(2 12)(3 9)(4 10)(5 27)(6 28)(7 25)(8 26)(13 29)(14 30)(15 31)(16 32)(17 21)(18 22)(19 23)(20 24)(33 43)(34 44)(35 41)(36 42)(37 47)(38 48)(39 45)(40 46)

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

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

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

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++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6D4C3xD42+ 1+4C3x2+ 1+4
kernelC3xC22.29C24C3xC42:C2C3xC22wrC2C3xC4:D4C3xC4.4D4C3xC4:1D4D4xC2xC6C6xC4oD4C22.29C24C42:C2C22wrC2C4:D4C4.4D4C4:1D4C22xD4C2xC4oD4C2xC12C2xC4C6C2
# reps11442211228844224824

Matrix representation of C3xC22.29C24 in 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] >;

C3xC22.29C24 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

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