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G = C3xD4oD8order 192 = 26·3

Direct product of C3 and D4oD8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3xD4oD8, C24.48C23, C12.85C24, 2+ 1+4:8C6, C8oD4:7C6, C4oD8:4C6, D8:7(C2xC6), (C2xD8):12C6, (C6xD8):26C2, C8:C22:4C6, Q16:7(C2xC6), C4.45(C6xD4), SD16:4(C2xC6), D4.11(C3xD4), (C3xD4).45D4, C4.8(C23xC6), Q8.16(C3xD4), (C3xQ8).45D4, C22.7(C6xD4), (C2xC24):23C22, C12.406(C2xD4), (C6xD4):40C22, (C3xD8):21C22, M4(2):6(C2xC6), C8.10(C22xC6), D4.5(C22xC6), Q8.9(C22xC6), (C3xQ16):21C22, (C3xD4).38C23, C6.206(C22xD4), (C3xQ8).39C23, (C2xC12).687C23, (C3xSD16):20C22, (C3x2+ 1+4):9C2, (C3xM4(2)):27C22, (C2xC8):4(C2xC6), C2.30(D4xC2xC6), C4oD4:2(C2xC6), (C3xC8oD4):8C2, (C2xD4):7(C2xC6), (C3xC4oD8):11C2, (C3xC8:C22):11C2, (C2xC6).184(C2xD4), (C3xC4oD4):14C22, (C2xC4).48(C22xC6), SmallGroup(192,1465)

Series: Derived Chief Lower central Upper central

C1C4 — C3xD4oD8
C1C2C4C12C3xD4C3xD8C6xD8 — C3xD4oD8
C1C2C4 — C3xD4oD8
C1C6C3xC4oD4 — C3xD4oD8

Generators and relations for C3xD4oD8
 G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C12, C12, C12, C2xC6, C2xC6, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xD4, C4oD4, C4oD4, C4oD4, C24, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xC6, C8oD4, C2xD8, C4oD8, C8:C22, 2+ 1+4, C2xC24, C3xM4(2), C3xD8, C3xSD16, C3xQ16, C6xD4, C6xD4, C3xC4oD4, C3xC4oD4, C3xC4oD4, D4oD8, C3xC8oD4, C6xD8, C3xC4oD8, C3xC8:C22, C3x2+ 1+4, C3xD4oD8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C24, C3xD4, C22xC6, C22xD4, C6xD4, C23xC6, D4oD8, D4xC2xC6, C3xD4oD8

Smallest permutation representation of C3xD4oD8
On 48 points
Generators in S48
(1 35 19)(2 36 20)(3 37 21)(4 38 22)(5 39 23)(6 40 24)(7 33 17)(8 34 18)(9 41 26)(10 42 27)(11 43 28)(12 44 29)(13 45 30)(14 46 31)(15 47 32)(16 48 25)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 34 29 38)(26 35 30 39)(27 36 31 40)(28 37 32 33)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(33 37)(34 36)(38 40)(42 48)(43 47)(44 46)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E···2J3A3B4A4B4C4D4E4F6A6B6C···6H6I···6T8A8B8C8D8E12A···12H12I12J12K12L24A24B24C24D24E···24J
order122222···233444444666···66···68888812···12121212122424242424···24
size112224···411222244112···24···4224442···2444422224···4

66 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3xD4C3xD4D4oD8C3xD4oD8
kernelC3xD4oD8C3xC8oD4C6xD8C3xC4oD8C3xC8:C22C3x2+ 1+4D4oD8C8oD4C2xD8C4oD8C8:C222+ 1+4C3xD4C3xQ8D4Q8C3C1
# reps1133622266124316224

Matrix representation of C3xD4oD8 in GL4(F7) generated by

4000
0400
0040
0004
,
6635
3651
1001
1132
,
6014
0545
0455
0665
,
5051
1521
1625
5511
,
0450
4610
4520
6146
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,3,1,1,6,6,0,1,3,5,0,3,5,1,1,2],[6,0,0,0,0,5,4,6,1,4,5,6,4,5,5,5],[5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[0,4,4,6,4,6,5,1,5,1,2,4,0,0,0,6] >;

C3xD4oD8 in GAP, Magma, Sage, TeX

C_3\times D_4\circ D_8
% in TeX

G:=Group("C3xD4oD8");
// GroupNames label

G:=SmallGroup(192,1465);
// by ID

G=gap.SmallGroup(192,1465);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,745,6053,3036,124]);
// Polycyclic

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

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