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G = C8xD4order 64 = 26

Direct product of C8 and D4

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

Aliases: C8xD4, C42.69C22, C8o(C4:C8), (C4xC8):4C2, C4:1(C2xC8), C8o2(C4:C4), C4:C8:18C2, C2.3(C4xD4), C4:C4.11C4, C8o(C22:C8), C22:1(C2xC8), (C22xC8):5C2, C4.76(C2xD4), C8o2(C22:C4), C22:C8:15C2, (C2xD4).11C4, (C4xD4).14C2, C2.2(C8oD4), C22:C4.7C4, C2.4(C22xC8), C4.51(C4oD4), (C2xC8).62C22, C23.18(C2xC4), (C2xC4).153C23, (C22xC4).95C22, C22.23(C22xC4), C4:C4o(C2xC8), (C2xC8)o(C2xD4), (C2xC8)o(C4xD4), (C2xC8)o(C4:C8), C22:C4o(C2xC8), (C2xC8)o(C22:C8), (C2xC4).35(C2xC4), SmallGroup(64,115)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8xD4
C1C2C4C2xC4C2xC8C22xC8 — C8xD4
C1C2 — C8xD4
C1C2xC8 — C8xD4
C1C2C2C2xC4 — C8xD4

Generators and relations for C8xD4
 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, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, C2xC8, C22xC4, C2xD4, C4xC8, C22:C8, C4:C8, C4xD4, C22xC8, C8xD4
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C23, C2xC8, C22xC4, C2xD4, C4oD4, C4xD4, C22xC8, C8oD4, C8xD4

Smallest permutation representation of C8xD4
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)]])

C8xD4 is a maximal subgroup of
D4:C16  SD16:C8  C8:9D8  C8:12SD16  D4.M4(2)  D4:2M4(2)  C8:8D8  C8:7D8  C8.28D8  C8:11SD16  C8:10SD16  D4.1Q16  D4.2SD16  D4.3SD16  D4.2D8  D4.Q16  C16:9D4  C16:6D4  C42.264C23  C42.681C23  M4(2):22D4  M4(2):23D4  C42.291C23  C42.293C23  C42.294C23  D4:6M4(2)  D4:7M4(2)  C42.297C23  C42.298C23  C42.694C23  C42.301C23  D4:8M4(2)  C42.307C23  C42.308C23  C42.309C23  D8:12D4  SD16:10D4  D8:13D4  SD16:11D4  Q16:12D4  Q16:13D4  D4:4D8  D4:7SD16  C42.461C23  C42.462C23  D4:8SD16  D4:5Q16  C42.465C23  C42.466C23  C42.467C23  C42.468C23  C42.469C23  C42.470C23  D4:5D8  D4:9SD16  C42.485C23  C42.486C23  D4:6Q16  C42.488C23  C42.489C23  C42.490C23  C42.491C23
 D4p:C8: D8:5C8  D12:C8  D20:5C8  D20:2C8  D28:C8 ...
 D2p:(C2xC8): C42.691C23  C42.697C23  C3:D4:C8  C5:5(C8xD4)  C5:C8:8D4  C7:D4:C8 ...
C8xD4 is a maximal quotient of
SD16:C8  Q16:5C8  C23.21M4(2)  C23.22M4(2)  C4:C4:3C8  C22:C4:4C8  C42.61Q8  C42.325D4
 D4p:C8: D8:5C8  D12:C8  D20:5C8  D20:2C8  D28:C8 ...
 C2p.(C4xD4): C16:9D4  C16:6D4  C16oD8  D8.C8  C3:D4:C8  C5:5(C8xD4)  C5:C8:8D4  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+++++++
imageC1C2C2C2C2C2C4C4C4C8D4C4oD4C8oD4
kernelC8xD4C4xC8C22:C8C4:C8C4xD4C22xC8C22:C4C4:C4C2xD4D4C8C4C2
# reps11211242216224

Matrix representation of C8xD4 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] >;

C8xD4 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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