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## G = C3×D4.10D4order 192 = 26·3

### Direct product of C3 and D4.10D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×D4.10D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×Q8 — C3×C8.C22 — C3×D4.10D4
 Lower central C1 — C2 — C2×C4 — C3×D4.10D4
 Upper central C1 — C6 — C2×C12 — C3×D4.10D4

Generators and relations for C3×D4.10D4
G = < a,b,c,d,e | a3=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 242 in 142 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C4×C12, C3×C4⋊C4, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, D4.10D4, C3×C4.10D4, C3×C4≀C2, C3×C4⋊Q8, C3×C8.C22, C3×2- 1+4, C3×D4.10D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, D4.10D4, C3×C22≀C2, C3×D4.10D4

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C ··· 4H 4I 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 12A 12B 12C 12D 12E ··· 12P 12Q 12R 24A 24B 24C 24D order 1 2 2 2 2 3 3 4 4 4 ··· 4 4 6 6 6 6 6 6 6 6 8 8 12 12 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 2 4 4 1 1 2 2 4 ··· 4 8 1 1 2 2 4 4 4 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8 8 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D4 C3×D4 C3×D4 C3×D4 D4.10D4 C3×D4.10D4 kernel C3×D4.10D4 C3×C4.10D4 C3×C4≀C2 C3×C4⋊Q8 C3×C8.C22 C3×2- 1+4 D4.10D4 C4.10D4 C4≀C2 C4⋊Q8 C8.C22 2- 1+4 C2×C12 C3×D4 C3×Q8 C2×C4 D4 Q8 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 2 2 4 4 4 2 4

Matrix representation of C3×D4.10D4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 2 3 1 0 1 1 0 4 6 5 5 2 1 4 5 6
,
 2 3 2 6 6 5 4 5 2 6 5 5 4 5 5 2
,
 4 3 3 6 2 4 1 4 0 5 5 6 1 1 1 1
,
 0 1 5 0 6 4 4 2 0 2 2 1 2 2 1 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,1,6,1,3,1,5,4,1,0,5,5,0,4,2,6],[2,6,2,4,3,5,6,5,2,4,5,5,6,5,5,2],[4,2,0,1,3,4,5,1,3,1,5,1,6,4,6,1],[0,6,0,2,1,4,2,2,5,4,2,1,0,2,1,1] >;

C3×D4.10D4 in GAP, Magma, Sage, TeX

C_3\times D_4._{10}D_4
% in TeX

G:=Group("C3xD4.10D4");
// GroupNames label

G:=SmallGroup(192,889);
// by ID

G=gap.SmallGroup(192,889);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,520,4204,2111,1068,172,3036]);
// Polycyclic

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

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