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G = D4×D8order 128 = 27

Direct product of D4 and D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: D4×D8, C42.446C23, C4.1332+ 1+4, D427C2, C42(C2×D8), C2.63D42, (C4×D8)⋊9C2, (C8×D4)⋊7C2, D45(C2×D4), C813(C2×D4), C87D49C2, C222(C2×D8), C22⋊D87C2, C4⋊D811C2, C84D411C2, C4⋊C861C22, C4⋊C4.252D4, (C4×C8)⋊11C22, (C2×D8)⋊8C22, (C2×D4).347D4, C2.41(D4○D8), (C4×D4)⋊21C22, (C22×D8)⋊12C2, C22⋊C4.92D4, C4.93(C22×D4), C2.18(C22×D8), C2.D859C22, D4⋊C45C22, C41D412C22, C4⋊D412C22, C4⋊C4.218C23, C22⋊C854C22, (C2×C8).178C23, (C2×C4).477C24, (C22×C8)⋊11C22, C23.463(C2×D4), (C2×D4).416C23, (C22×D4)⋊28C22, C22.737(C22×D4), (C22×C4).1121C23, (C2×D4)(C2×D8), (C2×C4).160(C2×D4), SmallGroup(128,2011)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4×D8
C1C2C22C2×C4C2×D4C22×D4C22×D8 — D4×D8
C1C2C2×C4 — D4×D8
C1C22C4×D4 — D4×D8
C1C2C2C2×C4 — D4×D8

Generators and relations for D4×D8
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 824 in 310 conjugacy classes, 104 normal (24 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×40], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×8], D4 [×32], C23 [×2], C23 [×26], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×14], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×D4 [×32], C24 [×4], C4×C8, C22⋊C8 [×2], D4⋊C4 [×6], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C4⋊D4 [×2], C41D4 [×2], C22×C8 [×2], C2×D8, C2×D8 [×8], C2×D8 [×8], C22×D4 [×4], C22×D4 [×2], C8×D4, C4×D8, C22⋊D8 [×4], C4⋊D8 [×2], C87D4 [×2], C84D4, D42 [×2], C22×D8 [×2], D4×D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D8 [×4], C2×D4 [×12], C24, C2×D8 [×6], C22×D4 [×2], 2+ 1+4, D42, C22×D8, D4○D8, D4×D8

Smallest permutation representation of D4×D8
On 32 points
Generators in S32
(1 11 20 32)(2 12 21 25)(3 13 22 26)(4 14 23 27)(5 15 24 28)(6 16 17 29)(7 9 18 30)(8 10 19 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 19)(2 18)(3 17)(4 24)(5 23)(6 22)(7 21)(8 20)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E4F4G4H4I8A8B8C8D8E···8J
order122222222222222244444444488888···8
size111122224444888822224448822224···4

35 irreducible representations

dim1111111112222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4D82+ 1+4D4○D8
kernelD4×D8C8×D4C4×D8C22⋊D8C4⋊D8C87D4C84D4D42C22×D8C22⋊C4C4⋊C4D8C2×D4D4C4C2
# reps1114221222141812

Matrix representation of D4×D8 in GL4(𝔽17) generated by

16000
01600
00142
00123
,
1000
0100
00142
00133
,
14300
141400
00160
00016
,
14300
3300
0010
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,12,0,0,2,3],[1,0,0,0,0,1,0,0,0,0,14,13,0,0,2,3],[14,14,0,0,3,14,0,0,0,0,16,0,0,0,0,16],[14,3,0,0,3,3,0,0,0,0,1,0,0,0,0,1] >;

D4×D8 in GAP, Magma, Sage, TeX

D_4\times D_8
% in TeX

G:=Group("D4xD8");
// GroupNames label

G:=SmallGroup(128,2011);
// by ID

G=gap.SmallGroup(128,2011);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,346,2804,1411,375,172]);
// Polycyclic

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

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