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G = C8×D4order 64 = 26

Direct product of C8 and D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C8×D4, C42.69C22, C8(C4⋊C8), (C4×C8)⋊4C2, C41(C2×C8), C82(C4⋊C4), C4⋊C818C2, C2.3(C4×D4), C4⋊C4.11C4, C8(C22⋊C8), C221(C2×C8), (C22×C8)⋊5C2, C4.76(C2×D4), C82(C22⋊C4), C22⋊C815C2, (C2×D4).11C4, (C4×D4).14C2, C2.2(C8○D4), C22⋊C4.7C4, C2.4(C22×C8), C4.51(C4○D4), (C2×C8).62C22, C23.18(C2×C4), (C2×C4).153C23, (C22×C4).95C22, C22.23(C22×C4), C4⋊C4(C2×C8), (C2×C8)(C2×D4), (C2×C8)(C4×D4), (C2×C8)(C4⋊C8), C22⋊C4(C2×C8), (C2×C8)(C22⋊C8), (C2×C4).35(C2×C4), SmallGroup(64,115)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8×D4
C1C2C4C2×C4C2×C8C22×C8 — C8×D4
C1C2 — C8×D4
C1C2×C8 — C8×D4
C1C2C2C2×C4 — C8×D4

Generators and relations for C8×D4
 G = < a,b,c | a8=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 89 in 67 conjugacy classes, 45 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×4], C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], C22×C4 [×2], C2×D4, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C8×D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8×D4

Smallest permutation representation of C8×D4
On 32 points
Generators in S32
(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)
(1 15 19 30)(2 16 20 31)(3 9 21 32)(4 10 22 25)(5 11 23 26)(6 12 24 27)(7 13 17 28)(8 14 18 29)
(1 5)(2 6)(3 7)(4 8)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 21)(18 22)(19 23)(20 24)

G:=sub<Sym(32)| (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), (1,15,19,30)(2,16,20,31)(3,9,21,32)(4,10,22,25)(5,11,23,26)(6,12,24,27)(7,13,17,28)(8,14,18,29), (1,5)(2,6)(3,7)(4,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,21)(18,22)(19,23)(20,24)>;

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), (1,15,19,30)(2,16,20,31)(3,9,21,32)(4,10,22,25)(5,11,23,26)(6,12,24,27)(7,13,17,28)(8,14,18,29), (1,5)(2,6)(3,7)(4,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,21)(18,22)(19,23)(20,24) );

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)], [(1,15,19,30),(2,16,20,31),(3,9,21,32),(4,10,22,25),(5,11,23,26),(6,12,24,27),(7,13,17,28),(8,14,18,29)], [(1,5),(2,6),(3,7),(4,8),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(17,21),(18,22),(19,23),(20,24)])

C8×D4 is a maximal subgroup of
D4⋊C16  SD16⋊C8  C89D8  C812SD16  D4.M4(2)  D42M4(2)  C88D8  C87D8  C8.28D8  C811SD16  C810SD16  D4.1Q16  D4.2SD16  D4.3SD16  D4.2D8  D4.Q16  C169D4  C166D4  C42.264C23  C42.681C23  M4(2)⋊22D4  M4(2)⋊23D4  C42.291C23  C42.293C23  C42.294C23  D46M4(2)  D47M4(2)  C42.297C23  C42.298C23  C42.694C23  C42.301C23  D48M4(2)  C42.307C23  C42.308C23  C42.309C23  D812D4  SD1610D4  D813D4  SD1611D4  Q1612D4  Q1613D4  D44D8  D47SD16  C42.461C23  C42.462C23  D48SD16  D45Q16  C42.465C23  C42.466C23  C42.467C23  C42.468C23  C42.469C23  C42.470C23  D45D8  D49SD16  C42.485C23  C42.486C23  D46Q16  C42.488C23  C42.489C23  C42.490C23  C42.491C23
 D4p⋊C8: D85C8  D12⋊C8  D205C8  D202C8  D28⋊C8 ...
 D2p⋊(C2×C8): C42.691C23  C42.697C23  C3⋊D4⋊C8  C55(C8×D4)  C5⋊C88D4  C7⋊D4⋊C8 ...
C8×D4 is a maximal quotient of
SD16⋊C8  Q165C8  C23.21M4(2)  C23.22M4(2)  C4⋊C43C8  C22⋊C44C8  C42.61Q8  C42.325D4
 D4p⋊C8: D85C8  D12⋊C8  D205C8  D202C8  D28⋊C8 ...
 C2p.(C4×D4): C169D4  C166D4  C16○D8  D8.C8  C3⋊D4⋊C8  C55(C8×D4)  C5⋊C88D4  C7⋊D4⋊C8 ...

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4L8A···8H8I···8T
order1222222244444···48···88···8
size1111222211112···21···12···2

40 irreducible representations

dim1111111111222
type+++++++
imageC1C2C2C2C2C2C4C4C4C8D4C4○D4C8○D4
kernelC8×D4C4×C8C22⋊C8C4⋊C8C4×D4C22×C8C22⋊C4C4⋊C4C2×D4D4C8C4C2
# reps11211242216224

Matrix representation of C8×D4 in GL3(𝔽17) generated by

1500
010
001
,
1600
0016
010
,
100
010
0016
G:=sub<GL(3,GF(17))| [15,0,0,0,1,0,0,0,1],[16,0,0,0,0,1,0,16,0],[1,0,0,0,1,0,0,0,16] >;

C8×D4 in GAP, Magma, Sage, TeX

C_8\times D_4
% in TeX

G:=Group("C8xD4");
// GroupNames label

G:=SmallGroup(64,115);
// by ID

G=gap.SmallGroup(64,115);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,86,88]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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