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G = C3×C24⋊C22  order 192 = 26·3

Direct product of C3 and C24⋊C22

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

Aliases: C3×C24⋊C22, C6.1702+ 1+4, C24⋊7(C2×C6), C42⋊15(C2×C6), C22≀C2⋊10C6, (C4×C12)⋊46C22, C4.4D4⋊15C6, (C23×C6)⋊5C22, (C6×Q8)⋊31C22, (C2×C6).381C24, (C2×C12).682C23, (C6×D4).223C22, C22.55(C23×C6), C23.24(C22×C6), (C22×C6).107C23, C2.22(C3×2+ 1+4), (C2×Q8)⋊7(C2×C6), C22⋊C4⋊8(C2×C6), (C2×D4).36(C2×C6), (C3×C22≀C2)⋊18C2, (C3×C4.4D4)⋊35C2, (C2×C4).41(C22×C6), (C3×C22⋊C4)⋊43C22, SmallGroup(192,1450)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C24⋊C22
Chief series
C1 — C2 — C22 — C2×C6 — C2×C12 — C6×Q8 — C3×C4.4D4 — C3×C24⋊C22
Lower central
C1 — C22 — C3×C24⋊C22
Upper central
C1 — C2×C6 — C3×C24⋊C22

Generators and relations for C3×C24⋊C22
 G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fbf=be=eb, gbg=bde, gcg=cd=dc, ce=ec, fcf=cde, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 506 in 260 conjugacy classes, 142 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, D4, Q8, C23, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C24, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22≀C2, C4.4D4, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C23×C6, C24⋊C22, C3×C22≀C2, C3×C4.4D4, C3×C24⋊C22
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2+ 1+4, C23×C6, C24⋊C22, C3×2+ 1+4, C3×C24⋊C22

Smallest permutation representation of C3×C24⋊C22
►On 48 points
Generators in S48
(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)

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

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

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

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

(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)(13 16)(14 17)(15 18)(19 26)(20 27)(21 25)(22 30)(23 28)(24 29)(43 48)(44 46)(45 47)

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

57 conjugacy classes

class 1 2A2B2C2D···2I3A3B4A···4I6A···6F6G···6R12A···12R
order12222···2334···46···66···612···12
size11114···4114···41···14···44···4

57 irreducible representations

dim11111144
type++++
imageC1C2C2C3C6C62+ 1+4C3×2+ 1+4
kernelC3×C24⋊C22C3×C22≀C2C3×C4.4D4C24⋊C22C22≀C2C4.4D4C6C2
# reps1692121836

Matrix representation of C3×C24⋊C22 ►in GL9(𝔽13)

900000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
1200000000
000100000
0000120000
010000000
0012000000
000000010
0000000012
000001000
0000001200
,
100000000
0012000000
0120000000
000010000
000100000
0000001200
0000012000
000000001
000000010
,
100000000
010000000
001000000
000100000
000010000
0000012000
0000001200
0000000120
0000000012
,
100000000
0120000000
0012000000
0001200000
0000120000
0000012000
0000001200
0000000120
0000000012
,
100000000
0120000000
001000000
000100000
0000120000
0000012000
0000001200
000000010
000000001
,
1200000000
0120000000
0012000000
000100000
000010000
0000012000
000000100
0000000120
000000001

G:=sub<GL(9,GF(13))| [9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0],[1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1] >;
 

C3×C24⋊C22 in GAP, Magma, Sage, TeX 

C_3\times C_2^4\rtimes C_2^2
 
% in TeX
 
G:=Group("C3xC2^4:C2^2");
 
// GroupNames label
 
G:=SmallGroup(192,1450);
 
// by ID
 
G=gap.SmallGroup(192,1450);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,344,2102,1563,4259,794]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=e^2=f^2=g^2=1,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,b*d=d*b,f*b*f=b*e=e*b,g*b*g=b*d*e,g*c*g=c*d=d*c,c*e=e*c,f*c*f=c*d*e,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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