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G = C3×C22.11C24  order 192 = 26·3

Direct product of C3 and C22.11C24

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

Aliases: C3×C22.11C24, C6.1492+ (1+4), (C4×D4)⋊5C6, D4⋊7(C2×C12), (C6×D4)⋊23C4, C42⋊7(C2×C6), (D4×C12)⋊34C2, (C2×D4)⋊11C12, C23⋊3(C2×C12), (C4×C12)⋊38C22, C42⋊C2⋊6C6, C24.12(C2×C6), C6.59(C23×C4), C2.7(C23×C12), (C2×C6).338C24, C4.19(C22×C12), (C22×C12)⋊5C22, (C22×D4).10C6, C12.164(C22×C4), (C2×C12).709C23, (C6×D4).332C22, C22.11(C23×C6), C23.34(C22×C6), C22.2(C22×C12), (C23×C6).11C22, C2.1(C3×2+ (1+4)), (C22×C6).254C23, C4⋊C4⋊20(C2×C6), (C2×C4)⋊4(C2×C12), (D4×C2×C6).22C2, (C2×C12)⋊25(C2×C4), (C3×D4)⋊27(C2×C4), (C2×C22⋊C4)⋊5C6, (C22×C4)⋊4(C2×C6), (C22×C6)⋊5(C2×C4), C22⋊C4⋊18(C2×C6), (C3×C4⋊C4)⋊77C22, (C6×C22⋊C4)⋊10C2, (C2×D4).78(C2×C6), (C2×C4).56(C22×C6), (C2×C6).33(C22×C4), (C3×C42⋊C2)⋊27C2, (C3×C22⋊C4)⋊72C22, SmallGroup(192,1407)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C3×C22.11C24
Chief series
C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — D4×C12 — C3×C22.11C24
Lower central
C1 — C2 — C3×C22.11C24
Upper central
C1 — C2×C6 — C3×C22.11C24

Subgroups: 514 in 338 conjugacy classes, 242 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C6, C6 [×2], C6 [×10], C2×C4 [×14], C2×C4 [×8], D4 [×16], C23, C23 [×12], C23 [×4], C12 [×4], C12 [×8], C2×C6, C2×C6 [×10], C2×C6 [×18], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], C2×C12 [×14], C2×C12 [×8], C3×D4 [×16], C22×C6, C22×C6 [×12], C22×C6 [×4], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C4×C12 [×4], C3×C22⋊C4 [×12], C3×C4⋊C4 [×4], C22×C12, C22×C12 [×8], C6×D4 [×12], C23×C6 [×2], C22.11C24, C6×C22⋊C4 [×4], C3×C42⋊C2 [×2], D4×C12 [×8], D4×C2×C6, C3×C22.11C24

Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, C2×C12 [×28], C22×C6 [×15], C23×C4, 2+ (1+4) [×2], C22×C12 [×14], C23×C6, C22.11C24, C23×C12, C3×2+ (1+4) [×2], C3×C22.11C24

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽13)

30000
09000
00900
00090
00009
,
10000
012000
001200
000120
000012
,
120000
01000
00100
00010
00001
,
50000
00010
00001
01000
00100
,
120000
01000
001200
000120
00001
,
10000
00100
01000
00001
00010
,
10000
01000
00100
000120
000012

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12] >;
 

102 conjugacy classes

class 1 2A2B2C2D···2M3A3B4A···4T6A···6F6G···6Z12A···12AN
order12222···2334···46···66···612···12
size11112···2112···21···12···22···2

102 irreducible representations

dim11111111111144
type++++++
imageC1C2C2C2C2C3C4C6C6C6C6C122+ (1+4)C3×2+ (1+4)
kernelC3×C22.11C24C6×C22⋊C4C3×C42⋊C2D4×C12D4×C2×C6C22.11C24C6×D4C2×C22⋊C4C42⋊C2C4×D4C22×D4C2×D4C6C2
# reps14281216841623224

In GAP, Magma, Sage, TeX 

C_3\times C_2^2._{11}C_2^4
 
% in TeX
 
G:=Group("C3xC2^2.11C2^4");
 
// GroupNames label
 
G:=SmallGroup(192,1407);
 
// by ID
 
G=gap.SmallGroup(192,1407);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,555,1571]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=e^2=f^2=g^2=1,d^2=c,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=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*f=f*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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