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## G = C3×D4⋊5D4order 192 = 26·3

### Direct product of C3 and D4⋊5D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×D4⋊5D4
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C6×D4 — C3×C22≀C2 — C3×D4⋊5D4
 Lower central C1 — C22 — C3×D4⋊5D4
 Upper central C1 — C2×C6 — C3×D4⋊5D4

Generators and relations for C3×D45D4
G = < a,b,c,d,e | a3=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 570 in 334 conjugacy classes, 166 normal (62 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C23×C6, D45D4, C6×C22⋊C4, D4×C12, C3×C22≀C2, C3×C4⋊D4, C3×C4⋊D4, C3×C22⋊Q8, C3×C22.D4, C3×C4.4D4, D4×C2×C6, C6×C4○D4, C3×D45D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C24, C3×D4, C22×C6, C22×D4, C2×C4○D4, 2+ 1+4, C6×D4, C3×C4○D4, C23×C6, D45D4, D4×C2×C6, C6×C4○D4, C3×2+ 1+4, C3×D45D4

Smallest permutation representation of C3×D45D4
On 48 points
Generators in S48
(1 5 19)(2 6 20)(3 7 17)(4 8 18)(9 27 21)(10 28 22)(11 25 23)(12 26 24)(13 46 36)(14 47 33)(15 48 34)(16 45 35)(29 40 43)(30 37 44)(31 38 41)(32 39 42)
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 38)(39 40)(41 44)(42 43)(45 48)(46 47)
(1 34 11 32)(2 35 12 29)(3 36 9 30)(4 33 10 31)(5 15 25 39)(6 16 26 40)(7 13 27 37)(8 14 28 38)(17 46 21 44)(18 47 22 41)(19 48 23 42)(20 45 24 43)
(1 30)(2 31)(3 32)(4 29)(5 37)(6 38)(7 39)(8 40)(9 34)(10 35)(11 36)(12 33)(13 25)(14 26)(15 27)(16 28)(17 42)(18 43)(19 44)(20 41)(21 48)(22 45)(23 46)(24 47)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 3A 3B 4A ··· 4F 4G ··· 4L 6A ··· 6F 6G ··· 6R 6S ··· 6X 12A ··· 12L 12M ··· 12X order 1 2 2 2 2 ··· 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 ··· 2 4 4 4 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 2+ 1+4 C3×2+ 1+4 kernel C3×D4⋊5D4 C6×C22⋊C4 D4×C12 C3×C22≀C2 C3×C4⋊D4 C3×C22⋊Q8 C3×C22.D4 C3×C4.4D4 D4×C2×C6 C6×C4○D4 D4⋊5D4 C2×C22⋊C4 C4×D4 C22≀C2 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C22×D4 C2×C4○D4 C3×D4 C2×C6 D4 C22 C6 C2 # reps 1 2 2 2 3 1 2 1 1 1 2 4 4 4 6 2 4 2 2 2 4 4 8 8 1 2

Matrix representation of C3×D45D4 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 1 2 0 0 12 12
,
 12 0 0 0 0 12 0 0 0 0 1 2 0 0 0 12
,
 0 12 0 0 1 0 0 0 0 0 8 3 0 0 5 5
,
 0 12 0 0 12 0 0 0 0 0 5 10 0 0 8 8
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,1,12,0,0,2,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,2,12],[0,1,0,0,12,0,0,0,0,0,8,5,0,0,3,5],[0,12,0,0,12,0,0,0,0,0,5,8,0,0,10,8] >;

C3×D45D4 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_5D_4
% in TeX

G:=Group("C3xD4:5D4");
// GroupNames label

G:=SmallGroup(192,1435);
// by ID

G=gap.SmallGroup(192,1435);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,2102,794]);
// Polycyclic

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

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