Copied to
clipboard

G = C3xC12.48D4order 288 = 25·32

Direct product of C3 and C12.48D4

direct product, metabelian, supersoluble, monomial

Aliases: C3xC12.48D4, C62:9Q8, C62.192C23, C6.7(C6xQ8), C4:Dic3:8C6, (C2xC6):9Dic6, C6.39(C6xD4), Dic3:C4:2C6, (C2xDic6):6C6, C12.57(C3xD4), C2.9(C6xDic6), (C6xDic6):30C2, (C2xC12).351D6, (C3xC12).159D4, C23.30(S3xC6), C6.54(C2xDic6), C22:4(C3xDic6), (C22xC12).40S3, (C22xC12).19C6, C6.D4.4C6, (C22xC6).125D6, C6.124(C4oD12), C32:22(C22:Q8), C12.140(C3:D4), (C6xC12).324C22, (C2xC62).95C22, (C6xDic3).97C22, (C2xC6):5(C3xQ8), (C2xC6xC12).15C2, C2.5(C6xC3:D4), C3:4(C3xC22:Q8), (C2xC4).83(S3xC6), C6.14(C3xC4oD4), C4.23(C3xC3:D4), C22.54(S3xC2xC6), (C3xC6).50(C2xQ8), (C2xC12).91(C2xC6), (C3xC4:Dic3):32C2, C2.17(C3xC4oD12), (C3xC6).251(C2xD4), C6.140(C2xC3:D4), (C22xC4).9(C3xS3), (C3xDic3:C4):34C2, (C2xC6).47(C22xC6), (C22xC6).59(C2xC6), (C2xDic3).9(C2xC6), (C3xC6).102(C4oD4), (C2xC6).325(C22xS3), (C3xC6.D4).9C2, SmallGroup(288,695)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C3xC12.48D4
C1C3C6C2xC6C62C6xDic3C6xDic6 — C3xC12.48D4
C3C2xC6 — C3xC12.48D4
C1C2xC6C22xC12

Generators and relations for C3xC12.48D4
 G = < a,b,c,d | a3=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 346 in 175 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C2xC4, C2xC4, Q8, C23, C32, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xQ8, C3xC6, C3xC6, Dic6, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xC6, C22xC6, C22:Q8, C3xDic3, C3xC12, C3xC12, C62, C62, C62, Dic3:C4, C4:Dic3, C6.D4, C3xC22:C4, C3xC4:C4, C2xDic6, C22xC12, C22xC12, C6xQ8, C3xDic6, C6xDic3, C6xC12, C6xC12, C2xC62, C12.48D4, C3xC22:Q8, C3xDic3:C4, C3xC4:Dic3, C3xC6.D4, C6xDic6, C2xC6xC12, C3xC12.48D4
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2xC6, C2xD4, C2xQ8, C4oD4, C3xS3, Dic6, C3:D4, C3xD4, C3xQ8, C22xS3, C22xC6, C22:Q8, S3xC6, C2xDic6, C4oD12, C2xC3:D4, C6xD4, C6xQ8, C3xC4oD4, C3xDic6, C3xC3:D4, S3xC2xC6, C12.48D4, C3xC22:Q8, C6xDic6, C3xC4oD12, C6xC3:D4, C3xC12.48D4

Smallest permutation representation of C3xC12.48D4
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 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 28 14 43)(2 27 15 42)(3 26 16 41)(4 25 17 40)(5 36 18 39)(6 35 19 38)(7 34 20 37)(8 33 21 48)(9 32 22 47)(10 31 23 46)(11 30 24 45)(12 29 13 44)
(1 43 7 37)(2 42 8 48)(3 41 9 47)(4 40 10 46)(5 39 11 45)(6 38 12 44)(13 29 19 35)(14 28 20 34)(15 27 21 33)(16 26 22 32)(17 25 23 31)(18 36 24 30)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E4F4G4H6A···6F6G···6AE12A···12AF12AG···12AN
order12222233333444444446···66···612···1212···12
size111122112222222121212121···12···22···212···12

90 irreducible representations

dim111111111111222222222222222222
type++++++++-++-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4Q8D6D6C4oD4C3xS3C3:D4C3xD4Dic6C3xQ8S3xC6S3xC6C4oD12C3xC4oD4C3xC3:D4C3xDic6C3xC4oD12
kernelC3xC12.48D4C3xDic3:C4C3xC4:Dic3C3xC6.D4C6xDic6C2xC6xC12C12.48D4Dic3:C4C4:Dic3C6.D4C2xDic6C22xC12C22xC12C3xC12C62C2xC12C22xC6C3xC6C22xC4C12C12C2xC6C2xC6C2xC4C23C6C6C4C22C2
# reps121211242422122212244444244888

Matrix representation of C3xC12.48D4 in GL4(F13) generated by

3000
0300
0090
0009
,
6000
41100
00110
0006
,
11100
01200
00012
0010
,
12200
12100
0001
00120
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[6,4,0,0,0,11,0,0,0,0,11,0,0,0,0,6],[1,0,0,0,11,12,0,0,0,0,0,1,0,0,12,0],[12,12,0,0,2,1,0,0,0,0,0,12,0,0,1,0] >;

C3xC12.48D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{48}D_4
% in TeX

G:=Group("C3xC12.48D4");
// GroupNames label

G:=SmallGroup(288,695);
// by ID

G=gap.SmallGroup(288,695);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,590,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<