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G = C3×C243C4order 192 = 26·3

Direct product of C3 and C243C4

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

Aliases: C3×C243C4, C247C12, C25.5C6, (C23×C6)⋊5C4, (C24×C6).1C2, C6.87C22≀C2, C24.29(C2×C6), C23.38(C3×D4), C22.29(C6×D4), C23.29(C2×C12), (C22×C12)⋊3C22, (C22×C6).153D4, (C23×C6).83C22, C23.56(C22×C6), (C22×C6).443C23, C22.28(C22×C12), (C2×C22⋊C4)⋊1C6, (C6×C22⋊C4)⋊5C2, (C22×C4)⋊2(C2×C6), C2.4(C6×C22⋊C4), (C2×C6)⋊7(C22⋊C4), C2.1(C3×C22≀C2), (C2×C6).596(C2×D4), C6.91(C2×C22⋊C4), C223(C3×C22⋊C4), (C22×C6).110(C2×C4), (C2×C6).215(C22×C4), SmallGroup(192,812)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C243C4
C1C2C22C23C22×C6C22×C12C6×C22⋊C4 — C3×C243C4
C1C22 — C3×C243C4
C1C22×C6 — C3×C243C4

Generators and relations for C3×C243C4
 G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 898 in 506 conjugacy classes, 130 normal (10 characteristic)
C1, C2 [×7], C2 [×12], C3, C4 [×4], C22, C22 [×18], C22 [×68], C6 [×7], C6 [×12], C2×C4 [×12], C23, C23 [×18], C23 [×68], C12 [×4], C2×C6, C2×C6 [×18], C2×C6 [×68], C22⋊C4 [×12], C22×C4 [×4], C24 [×7], C24 [×12], C2×C12 [×12], C22×C6, C22×C6 [×18], C22×C6 [×68], C2×C22⋊C4 [×6], C25, C3×C22⋊C4 [×12], C22×C12 [×4], C23×C6 [×7], C23×C6 [×12], C243C4, C6×C22⋊C4 [×6], C24×C6, C3×C243C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×12], C23, C12 [×4], C2×C6 [×7], C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C12 [×6], C3×D4 [×12], C22×C6, C2×C22⋊C4 [×3], C22≀C2 [×4], C3×C22⋊C4 [×12], C22×C12, C6×D4 [×6], C243C4, C6×C22⋊C4 [×3], C3×C22≀C2 [×4], C3×C243C4

Smallest permutation representation of C3×C243C4
On 48 points
Generators in S48
(1 31 35)(2 32 36)(3 29 33)(4 30 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(2 12)(4 10)(5 30)(7 32)(14 34)(16 36)(18 38)(20 40)(22 42)(24 44)(26 48)(28 46)
(2 40)(4 38)(5 28)(7 26)(10 18)(12 20)(14 22)(16 24)(30 46)(32 48)(34 42)(36 44)
(1 11)(2 12)(3 9)(4 10)(5 30)(6 31)(7 32)(8 29)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 47)(26 48)(27 45)(28 46)
(1 39)(2 40)(3 37)(4 38)(5 28)(6 25)(7 26)(8 27)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(29 45)(30 46)(31 47)(32 48)(33 41)(34 42)(35 43)(36 44)
(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)

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

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

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

84 conjugacy classes

class 1 2A···2G2H···2S3A3B4A···4H6A···6N6O···6AL12A···12P
order12···22···2334···46···66···612···12
size11···12···2114···41···12···24···4

84 irreducible representations

dim1111111122
type++++
imageC1C2C2C3C4C6C6C12D4C3×D4
kernelC3×C243C4C6×C22⋊C4C24×C6C243C4C23×C6C2×C22⋊C4C25C24C22×C6C23
# reps16128122161224

Matrix representation of C3×C243C4 in GL6(𝔽13)

100000
010000
009000
000900
000090
000009
,
100000
0120000
001000
0001200
000010
000001
,
100000
010000
001000
000100
000010
0000112
,
1200000
0120000
0012000
0001200
000010
000001
,
100000
010000
001000
000100
0000120
0000012
,
010000
100000
000100
0012000
0000111
0000012

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,12] >;

C3×C243C4 in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes_3C_4
% in TeX

G:=Group("C3xC2^4:3C4");
// GroupNames label

G:=SmallGroup(192,812);
// by ID

G=gap.SmallGroup(192,812);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations

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