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G = C2xS3xC4oD4order 192 = 26·3

Direct product of C2, S3 and C4oD4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2xS3xC4oD4, C6.11C25, D12:11C23, C12.46C24, D6.13C24, Dic6:11C23, Dic3.6C24, (C2xD4):49D6, (C2xQ8):41D6, (C4xS3):7C23, (C2xC12):6C23, D4:8(C22xS3), (C3xD4):9C23, (C22xC4):43D6, C3:D4:4C23, (C2xC6).2C24, Q8:8(C22xS3), (C3xQ8):8C23, (S3xD4):16C22, (C6xD4):52C22, C4.77(S3xC23), C2.12(S3xC24), (S3xQ8):19C22, (C6xQ8):45C22, C4oD12:24C22, (C2xD12):63C22, C22.2(S3xC23), D4:2S3:18C22, (C22xC12):27C22, Q8:3S3:18C22, (C2xDic3):12C23, (C2xDic6):74C22, C23.223(C22xS3), (C22xC6).247C23, (S3xC23).119C22, (C22xS3).249C23, (C22xDic3):53C22, (C2xS3xD4):30C2, C6:4(C2xC4oD4), (C2xS3xQ8):23C2, C3:4(C22xC4oD4), (C6xC4oD4):13C2, (S3xC2xC4):61C22, (S3xC22xC4):11C2, (C2xC4):9(C22xS3), (C2xC4oD12):36C2, (C2xD4:2S3):32C2, (C2xQ8:3S3):23C2, (C3xC4oD4):19C22, (C2xC3:D4):53C22, SmallGroup(192,1520)

Series: Derived Chief Lower central Upper central

C1C6 — C2xS3xC4oD4
C1C3C6D6C22xS3S3xC23S3xC22xC4 — C2xS3xC4oD4
C3C6 — C2xS3xC4oD4
C1C2xC4C2xC4oD4

Generators and relations for C2xS3xC4oD4
 G = < a,b,c,d,e,f | a2=b3=c2=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >

Subgroups: 1832 in 890 conjugacy classes, 455 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4oD4, C24, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xS3, C22xS3, C22xC6, C23xC4, C22xD4, C22xQ8, C2xC4oD4, C2xC4oD4, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, C4oD12, S3xD4, D4:2S3, S3xQ8, Q8:3S3, C22xDic3, C2xC3:D4, C22xC12, C6xD4, C6xQ8, C3xC4oD4, S3xC23, C22xC4oD4, S3xC22xC4, C2xC4oD12, C2xS3xD4, C2xD4:2S3, C2xS3xQ8, C2xQ8:3S3, S3xC4oD4, C6xC4oD4, C2xS3xC4oD4
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, C25, S3xC23, C22xC4oD4, S3xC4oD4, S3xC24, C2xS3xC4oD4

Smallest permutation representation of C2xS3xC4oD4
On 48 points
Generators in S48
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 39 44)(34 40 41)(35 37 42)(36 38 43)
(1 36)(2 33)(3 34)(4 35)(5 32)(6 29)(7 30)(8 31)(9 26)(10 27)(11 28)(12 25)(13 44)(14 41)(15 42)(16 43)(17 38)(18 39)(19 40)(20 37)(21 47)(22 48)(23 45)(24 46)
(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 46 3 48)(2 47 4 45)(5 15 7 13)(6 16 8 14)(9 19 11 17)(10 20 12 18)(21 35 23 33)(22 36 24 34)(25 39 27 37)(26 40 28 38)(29 43 31 41)(30 44 32 42)
(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(45 47)(46 48)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M2N···2S 3 4A4B4C4D4E···4J4K4L4M4N4O···4T6A6B6C6D···6I12A12B12C12D12E···12J
order12222···222222···2344444···444444···46666···61212121212···12
size11112···233336···6211112···233336···62224···422224···4

60 irreducible representations

dim1111111112222224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D6D6D6D6C4oD4S3xC4oD4
kernelC2xS3xC4oD4S3xC22xC4C2xC4oD12C2xS3xD4C2xD4:2S3C2xS3xQ8C2xQ8:3S3S3xC4oD4C6xC4oD4C2xC4oD4C22xC4C2xD4C2xQ8C4oD4D6C2
# reps13333111611331884

Matrix representation of C2xS3xC4oD4 in GL4(F13) generated by

12000
01200
00120
00012
,
121200
1000
0010
0001
,
12000
1100
0010
0001
,
12000
01200
0050
0005
,
12000
01200
00111
00112
,
1000
0100
0010
00112
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,1,1,0,0,11,12],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;

C2xS3xC4oD4 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_4\circ D_4
% in TeX

G:=Group("C2xS3xC4oD4");
// GroupNames label

G:=SmallGroup(192,1520);
// by ID

G=gap.SmallGroup(192,1520);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,136,438,6278]);
// Polycyclic

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

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