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## G = C3×C4⋊1D4order 96 = 25·3

### Direct product of C3 and C4⋊1D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C4⋊1D4
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊1D4
 Lower central C1 — C22 — C3×C4⋊1D4
 Upper central C1 — C2×C6 — C3×C4⋊1D4

Generators and relations for C3×C41D4
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×12], C6 [×3], C6 [×4], C2×C4 [×3], D4 [×12], C23 [×4], C12 [×6], C2×C6, C2×C6 [×12], C42, C2×D4 [×6], C2×C12 [×3], C3×D4 [×12], C22×C6 [×4], C41D4, C4×C12, C6×D4 [×6], C3×C41D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, C2×C6 [×7], C2×D4 [×3], C3×D4 [×6], C22×C6, C41D4, C6×D4 [×3], C3×C41D4

Smallest permutation representation of C3×C41D4
On 48 points
Generators in S48
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 34 42)(6 35 43)(7 36 44)(8 33 41)(9 37 46)(10 38 47)(11 39 48)(12 40 45)(17 22 30)(18 23 31)(19 24 32)(20 21 29)
(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 31 10 35)(2 32 11 36)(3 29 12 33)(4 30 9 34)(5 13 22 46)(6 14 23 47)(7 15 24 48)(8 16 21 45)(17 37 42 25)(18 38 43 26)(19 39 44 27)(20 40 41 28)
(1 30)(2 29)(3 32)(4 31)(5 47)(6 46)(7 45)(8 48)(9 35)(10 34)(11 33)(12 36)(13 23)(14 22)(15 21)(16 24)(17 26)(18 25)(19 28)(20 27)(37 43)(38 42)(39 41)(40 44)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4F 6A ··· 6F 6G ··· 6N 12A ··· 12L order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 4 4 4 4 1 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C3 C6 C6 D4 C3×D4 kernel C3×C4⋊1D4 C4×C12 C6×D4 C4⋊1D4 C42 C2×D4 C12 C4 # reps 1 1 6 2 2 12 6 12

Matrix representation of C3×C41D4 in GL4(𝔽13) generated by

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

C3×C41D4 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes_1D_4
% in TeX

G:=Group("C3xC4:1D4");
// GroupNames label

G:=SmallGroup(96,174);
// by ID

G=gap.SmallGroup(96,174);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,151,938,230]);
// Polycyclic

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

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