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G = D4×C12order 96 = 25·3

Direct product of C12 and D4

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

Aliases: D4×C12, C427C6, C4⋊C47C6, C41(C2×C12), C126(C2×C4), C122(C4⋊C4), C2.3(C6×D4), (C4×C12)⋊11C2, C22⋊C46C6, (C22×C4)⋊4C6, (C2×D4).7C6, C6.66(C2×D4), C222(C2×C12), (C22×C12)⋊4C2, (C6×D4).14C2, C122(C22⋊C4), C6.39(C4○D4), (C2×C6).73C23, C2.4(C22×C12), C23.10(C2×C6), C6.32(C22×C4), C22.7(C22×C6), (C2×C12).121C22, (C22×C6).26C22, C42(C3×C4⋊C4), C4⋊C4(C2×C12), (C2×C4)(C6×D4), C122(C3×C4⋊C4), (C2×C6)⋊4(C2×C4), (C2×C12)(C6×D4), (C2×D4)(C2×C12), (C3×C4⋊C4)⋊16C2, C22⋊C4(C2×C12), C42(C3×C22⋊C4), C2.2(C3×C4○D4), C122(C3×C22⋊C4), (C2×C4).15(C2×C6), (C3×C22⋊C4)⋊14C2, (C2×C4)(C3×C4⋊C4), (C2×C12)(C3×C4⋊C4), (C2×C4)(C3×C22⋊C4), (C2×C12)(C3×C22⋊C4), SmallGroup(96,165)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C12
C1C2C22C2×C6C2×C12C3×C22⋊C4 — D4×C12
C1C2 — D4×C12
C1C2×C12 — D4×C12

Generators and relations for D4×C12
 G = < a,b,c | a12=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C12 [×4], C12 [×3], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×C6 [×2], C4×D4, C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12 [×2], C6×D4, D4×C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×D4, C4○D4, C2×C12 [×6], C3×D4 [×2], C22×C6, C4×D4, C22×C12, C6×D4, C3×C4○D4, D4×C12

Smallest permutation representation of D4×C12
On 48 points
Generators in S48
(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 38 15 34)(2 39 16 35)(3 40 17 36)(4 41 18 25)(5 42 19 26)(6 43 20 27)(7 44 21 28)(8 45 22 29)(9 46 23 30)(10 47 24 31)(11 48 13 32)(12 37 14 33)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)

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

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

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

D4×C12 is a maximal subgroup of
C12.57D8  C12.50D8  C12.38SD16  D4.3Dic6  C42.47D6  C123M4(2)  C42.48D6  C127D8  D4.1D12  C42.51D6  D4.2D12  C42.102D6  D45Dic6  C42.104D6  C42.105D6  C42.106D6  D46Dic6  C4213D6  C42.108D6  C4214D6  C42.228D6  D1223D4  D1224D4  Dic623D4  Dic624D4  D45D12  D46D12  C4218D6  C42.229D6  C42.113D6  C42.114D6  C4219D6  C42.115D6  C42.116D6  C42.117D6  C42.118D6  C42.119D6

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E···4L6A···6F6G···6N12A···12H12I···12X
order122222223344444···46···66···612···1212···12
size111122221111112···21···12···21···12···2

60 irreducible representations

dim111111111111112222
type+++++++
imageC1C2C2C2C2C2C3C4C6C6C6C6C6C12D4C4○D4C3×D4C3×C4○D4
kernelD4×C12C4×C12C3×C22⋊C4C3×C4⋊C4C22×C12C6×D4C4×D4C3×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D4C12C6C4C2
# reps1121212824242162244

Matrix representation of D4×C12 in GL3(𝔽13) generated by

600
0100
0010
,
100
012
01212
,
100
010
01212
G:=sub<GL(3,GF(13))| [6,0,0,0,10,0,0,0,10],[1,0,0,0,1,12,0,2,12],[1,0,0,0,1,12,0,0,12] >;

D4×C12 in GAP, Magma, Sage, TeX

D_4\times C_{12}
% in TeX

G:=Group("D4xC12");
// GroupNames label

G:=SmallGroup(96,165);
// by ID

G=gap.SmallGroup(96,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,230]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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