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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

Aliases: C3×D4⋊C4, D41C12, C6.13D8, C12.60D4, C6.9SD16, C4⋊C41C6, (C2×C8)⋊2C6, (C2×C24)⋊4C2, (C3×D4)⋊4C4, C2.1(C3×D8), C4.1(C2×C12), (C2×D4).3C6, (C6×D4).9C2, C4.11(C3×D4), (C2×C6).46D4, C12.28(C2×C4), C2.1(C3×SD16), C22.8(C3×D4), C6.24(C22⋊C4), (C2×C12).114C22, (C3×C4⋊C4)⋊10C2, (C2×C4).17(C2×C6), C2.6(C3×C22⋊C4), SmallGroup(96,52)

Series: Derived Chief Lower central Upper central

C1C4 — C3×D4⋊C4
C1C2C22C2×C4C2×C12C3×C4⋊C4 — C3×D4⋊C4
C1C2C4 — C3×D4⋊C4
C1C2×C6C2×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 >

4C2
4C2
2C22
2C22
4C22
4C22
4C4
4C6
4C6
2C2×C4
2C23
2C8
2D4
2C2×C6
2C2×C6
4C2×C6
4C12
4C2×C6
2C24
2C22×C6
2C2×C12
2C3×D4

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)])

C3×D4⋊C4 is a maximal subgroup of
Dic34D8  D4.S3⋊C4  Dic36SD16  Dic3.D8  Dic3.SD16  D4⋊Dic6  Dic62D4  D4.Dic6  C4⋊C4.D6  C12⋊Q8⋊C2  D4.2Dic6  Dic6.D4  (C2×C8).200D6  C4⋊C419D6  D4⋊(C4×S3)  D42S3⋊C4  D4⋊D12  D6.D8  D6⋊D8  D65SD16  D6.SD16  D6⋊SD16  D6⋊C811C2  C3⋊C81D4  D43D12  C3⋊C8⋊D4  D4.D12  C241C4⋊C2  D4⋊S3⋊C4  D123D4  D12.D4  C12×D8  C12×SD16

42 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D6A···6F6G6H6I6J8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order1222223344446···666668888121212121212121224···24
size1111441122441···144442222222244442···2

42 irreducible representations

dim111111111122222222
type+++++++
imageC1C2C2C2C3C4C6C6C6C12D4D4D8SD16C3×D4C3×D4C3×D8C3×SD16
kernelC3×D4⋊C4C3×C4⋊C4C2×C24C6×D4D4⋊C4C3×D4C4⋊C4C2×C8C2×D4D4C12C2×C6C6C6C4C22C2C2
# reps111124222811222244

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

100
0640
0064
,
100
07272
021
,
100
0720
021
,
2700
0067
0610
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

Export

Subgroup lattice of C3×D4⋊C4 in TeX

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