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## G = C3×D4⋊C4order 96 = 25·3

### Direct product of C3 and D4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×D4⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×D4⋊C4
 Lower central C1 — C2 — C4 — C3×D4⋊C4
 Upper central C1 — C2×C6 — C2×C12 — C3×D4⋊C4

Generators and relations for C3×D4⋊C4
G = < a,b,c,d | a3=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Smallest permutation representation of C3×D4⋊C4
On 48 points
Generators in S48
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 42 34)(6 43 35)(7 44 36)(8 41 33)(9 28 17)(10 25 18)(11 26 19)(12 27 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 10)(2 9)(3 12)(4 11)(5 48)(6 47)(7 46)(8 45)(13 17)(14 20)(15 19)(16 18)(21 28)(22 27)(23 26)(24 25)(29 33)(30 36)(31 35)(32 34)(37 41)(38 44)(39 43)(40 42)
(1 36 11 32)(2 35 12 31)(3 34 9 30)(4 33 10 29)(5 28 46 22)(6 27 47 21)(7 26 48 24)(8 25 45 23)(13 43 20 39)(14 42 17 38)(15 41 18 37)(16 44 19 40)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D8 SD16 C3×D4 C3×D4 C3×D8 C3×SD16 kernel C3×D4⋊C4 C3×C4⋊C4 C2×C24 C6×D4 D4⋊C4 C3×D4 C4⋊C4 C2×C8 C2×D4 D4 C12 C2×C6 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 4 4

Matrix representation of C3×D4⋊C4 in GL3(𝔽73) generated by

 1 0 0 0 64 0 0 0 64
,
 1 0 0 0 72 72 0 2 1
,
 1 0 0 0 72 0 0 2 1
,
 27 0 0 0 0 67 0 61 0
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[1,0,0,0,72,2,0,72,1],[1,0,0,0,72,2,0,0,1],[27,0,0,0,0,61,0,67,0] >;

C3×D4⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes C_4
% in TeX

G:=Group("C3xD4:C4");
// GroupNames label

G:=SmallGroup(96,52);
// by ID

G=gap.SmallGroup(96,52);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,729,117]);
// Polycyclic

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

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