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G = C3×D4.4D4order 192 = 26·3

Direct product of C3 and D4.4D4

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

Aliases: C3×D4.4D4, C24.93D4, C8○D46C6, (C2×D8)⋊8C6, (C6×D8)⋊22C2, C8⋊C223C6, D4.4(C3×D4), C8.19(C3×D4), C4.38(C6×D4), Q8.9(C3×D4), C8.C46C6, (C3×D4).29D4, C4.D44C6, (C3×Q8).29D4, C12.399(C2×D4), M4(2).4(C2×C6), C6.155(C4⋊D4), (C2×C24).201C22, (C2×C12).614C23, (C6×D4).194C22, (C3×M4(2)).48C22, (C3×C8○D4)⋊7C2, (C2×C8).25(C2×C6), (C3×C8⋊C22)⋊10C2, C4○D4.18(C2×C6), (C2×D4).17(C2×C6), (C3×C4.D4)⋊8C2, C2.24(C3×C4⋊D4), (C2×C4).9(C22×C6), (C3×C8.C4)⋊15C2, C22.7(C3×C4○D4), (C2×C6).116(C4○D4), (C3×C4○D4).56C22, SmallGroup(192,905)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C3×D4.4D4
C1C2C4C2×C4C2×C12C6×D4C6×D8 — C3×D4.4D4
C1C2C2×C4 — C3×D4.4D4
C1C6C2×C12 — C3×D4.4D4

Generators and relations for C3×D4.4D4
 G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=b2d3 >

Subgroups: 226 in 108 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C3×M4(2), C3×D8, C3×SD16, C6×D4, C3×C4○D4, D4.4D4, C3×C4.D4, C3×C8.C4, C3×C8○D4, C6×D8, C3×C8⋊C22, C3×D4.4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C6×D4, C3×C4○D4, D4.4D4, C3×C4⋊D4, C3×D4.4D4

Smallest permutation representation of C3×D4.4D4
On 48 points
Generators in S48
(1 33 23)(2 34 24)(3 35 17)(4 36 18)(5 37 19)(6 38 20)(7 39 21)(8 40 22)(9 48 28)(10 41 29)(11 42 30)(12 43 31)(13 44 32)(14 45 25)(15 46 26)(16 47 27)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)
(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 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 40)(34 39)(35 38)(36 37)(41 42)(43 48)(44 47)(45 46)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C6A6B6C6D6E6F6G6H6I6J8A8B8C8D8E8F8G12A12B12C12D12E12F24A24B24C24D24E···24J24K24L24M24N
order12222233444666666666688888881212121212122424242424···2424242424
size112488112241122448888224448822224422224···48888

48 irreducible representations

dim1111111111112222222244
type++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D4C4○D4C3×D4C3×D4C3×D4C3×C4○D4D4.4D4C3×D4.4D4
kernelC3×D4.4D4C3×C4.D4C3×C8.C4C3×C8○D4C6×D8C3×C8⋊C22D4.4D4C4.D4C8.C4C8○D4C2×D8C8⋊C22C24C3×D4C3×Q8C2×C6C8D4Q8C22C3C1
# reps1211122422242112422424

Matrix representation of C3×D4.4D4 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
0651
3056
3361
1631
,
1253
4022
6602
6146
,
2026
6256
6152
2266
,
2425
6452
6015
2560
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[1,4,6,6,2,0,6,1,5,2,0,4,3,2,2,6],[2,6,6,2,0,2,1,2,2,5,5,6,6,6,2,6],[2,6,6,2,4,4,0,5,2,5,1,6,5,2,5,0] >;

C3×D4.4D4 in GAP, Magma, Sage, TeX

C_3\times D_4._4D_4
% in TeX

G:=Group("C3xD4.4D4");
// GroupNames label

G:=SmallGroup(192,905);
// by ID

G=gap.SmallGroup(192,905);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,4204,172,6053,1531,124]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=b^2*d^3>;
// generators/relations

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