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G = C3×C42⋊C2order 96 = 25·3

Direct product of C3 and C42⋊C2

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

Aliases: C3×C42⋊C2, C424C6, C4⋊C46C6, C12(C4⋊C4), (C2×C4)⋊4C12, (C4×C12)⋊2C2, (C2×C12)⋊9C4, C4.9(C2×C12), C12(C22⋊C4), C12.46(C2×C4), C22⋊C4.3C6, (C22×C4).6C6, C23.9(C2×C6), C6.38(C4○D4), C22.5(C2×C12), (C2×C6).72C23, C2.3(C22×C12), C6.31(C22×C4), (C22×C12).14C2, (C2×C12).79C22, C22.6(C22×C6), (C22×C6).25C22, C4(C3×C4⋊C4), C12(C3×C4⋊C4), C4(C3×C22⋊C4), (C3×C4⋊C4)⋊15C2, C12(C3×C22⋊C4), C2.1(C3×C4○D4), (C2×C6).22(C2×C4), (C2×C4).14(C2×C6), (C3×C22⋊C4).6C2, SmallGroup(96,164)

Series: Derived Chief Lower central Upper central

C1C2 — C3×C42⋊C2
C1C2C22C2×C6C2×C12C3×C22⋊C4 — C3×C42⋊C2
C1C2 — C3×C42⋊C2
C1C2×C12 — C3×C42⋊C2

Generators and relations for C3×C42⋊C2
 G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >

Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C23, C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×C12 [×2], C2×C12 [×8], C22×C6, C42⋊C2, C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×C12, C3×C42⋊C2
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C22×C4, C4○D4 [×2], C2×C12 [×6], C22×C6, C42⋊C2, C22×C12, C3×C4○D4 [×2], C3×C42⋊C2

Smallest permutation representation of C3×C42⋊C2
On 48 points
Generators in S48
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 27 45)(22 28 46)(23 25 47)(24 26 48)
(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 23 39 15)(2 24 40 16)(3 21 37 13)(4 22 38 14)(5 34 28 42)(6 35 25 43)(7 36 26 44)(8 33 27 41)(9 45 17 29)(10 46 18 30)(11 47 19 31)(12 48 20 32)
(1 3)(2 38)(4 40)(5 26)(6 8)(7 28)(9 11)(10 20)(12 18)(13 15)(14 24)(16 22)(17 19)(21 23)(25 27)(29 31)(30 48)(32 46)(33 35)(34 44)(36 42)(37 39)(41 43)(45 47)

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

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

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

C3×C42⋊C2 is a maximal subgroup of
C423Dic3  C12.2C42  (C2×C12).Q8  C4⋊C4.232D6  C4⋊C4.233D6  C12.5C42  C4⋊C4.234D6  C42.43D6  C42.187D6  C4⋊C436D6  C4.(C2×D12)  C4⋊C4.236D6  C4⋊C4.237D6  C426D6  (C2×D12)⋊13C4  C42.87D6  C42.88D6  C42.89D6  C42.90D6  C429D6  C42.188D6  C42.91D6  C4210D6  C4211D6  C42.92D6  C4212D6  C42.93D6  C42.94D6  C42.95D6  C42.96D6  C42.97D6  C42.98D6  C42.99D6  C42.100D6  C12×C4○D4
C3×C42⋊C2 is a maximal quotient of
C12×C22⋊C4  C12×C4⋊C4

60 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E···4N6A···6F6G6H6I6J12A···12H12I···12AB
order1222223344444···46···6666612···1212···12
size1111221111112···21···122221···12···2

60 irreducible representations

dim11111111111122
type+++++
imageC1C2C2C2C2C3C4C6C6C6C6C12C4○D4C3×C4○D4
kernelC3×C42⋊C2C4×C12C3×C22⋊C4C3×C4⋊C4C22×C12C42⋊C2C2×C12C42C22⋊C4C4⋊C4C22×C4C2×C4C6C2
# reps122212844421648

Matrix representation of C3×C42⋊C2 in GL3(𝔽13) generated by

100
030
003
,
500
0811
005
,
100
050
005
,
100
0120
051
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[5,0,0,0,8,0,0,11,5],[1,0,0,0,5,0,0,0,5],[1,0,0,0,12,5,0,0,1] >;

C3×C42⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes C_2
% in TeX

G:=Group("C3xC4^2:C2");
// GroupNames label

G:=SmallGroup(96,164);
// by ID

G=gap.SmallGroup(96,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,122]);
// 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*c^2,c*d=d*c>;
// generators/relations

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