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G = D4.D8order 128 = 27

1st non-split extension by D4 of D8 acting via D8/D4=C2

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

Aliases: D4.6D8, D43SD16, C42.200C23, D42.2C2, D4⋊C820C2, (C4×C8)⋊3C22, C4⋊C4.54D4, C4.28(C2×D8), C4⋊Q82C22, C4⋊C843C22, D4⋊Q81C2, C4.4D84C2, C4.D88C2, (C2×D4).254D4, C4.31(C2×SD16), D4.D430C2, C4.61(C8⋊C22), (C4×D4).31C22, C2.14(C22⋊D8), C41D4.17C22, C2.14(C22⋊SD16), C22.166C22≀C2, C2.12(D4.9D4), (C2×C4).957(C2×D4), SmallGroup(128,371)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — D4.D8
C1C2C22C2×C4C42C4×D4D42 — D4.D8
C1C22C42 — D4.D8
C1C22C42 — D4.D8
C1C22C22C42 — D4.D8

Generators and relations for D4.D8
 G = < a,b,c,d | a4=b2=c8=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 464 in 154 conjugacy classes, 38 normal (32 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×4], C22, C22 [×22], C8 [×3], C2×C4 [×3], C2×C4 [×7], D4 [×4], D4 [×16], Q8 [×2], C23 [×14], C42, C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×3], SD16 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×14], C2×Q8, C24 [×2], C4×C8, D4⋊C4 [×3], Q8⋊C4, C4⋊C8 [×2], C2.D8, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×2], C41D4, C4⋊Q8, C2×SD16, C22×D4 [×2], D4⋊C8 [×2], C4.D8, D4.D4, D4⋊Q8, C4.4D8, D42, D4.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], SD16 [×2], C2×D4 [×3], C22≀C2, C2×D8, C2×SD16, C8⋊C22 [×2], C22⋊D8, C22⋊SD16, D4.9D4, D4.D8

Character table of D4.D8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 111144448822224881644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-11-1-1-1111111-11-1-1-1-1-111-1    linear of order 2
ρ411111-11-1-1-1111111-1-111111-1-11    linear of order 2
ρ51111-1-1-1-11111111-1-1-1-1-1-1-11111    linear of order 2
ρ61111-1-1-1-11111111-1-111111-1-1-1-1    linear of order 2
ρ71111-11-11-1-111111-11-11111-111-1    linear of order 2
ρ81111-11-11-1-111111-111-1-1-1-11-1-11    linear of order 2
ρ922220000-22-2-2-2-2200000000000    orthogonal lifted from D4
ρ10222220200022-2-2-2-20000000000    orthogonal lifted from D4
ρ1122220-20-200-2-222-202000000000    orthogonal lifted from D4
ρ122222-20-200022-2-2-220000000000    orthogonal lifted from D4
ρ132222020200-2-222-20-2000000000    orthogonal lifted from D4
ρ14222200002-2-2-2-2-2200000000000    orthogonal lifted from D4
ρ1522-2-2020-20000-2200002-22-20-220    orthogonal lifted from D8
ρ1622-2-20-2020000-220000-22-220-220    orthogonal lifted from D8
ρ1722-2-2020-20000-220000-22-2202-20    orthogonal lifted from D8
ρ1822-2-20-2020000-2200002-22-202-20    orthogonal lifted from D8
ρ192-2-2220-20002-2000000-2--2--2-2--200-2    complex lifted from SD16
ρ202-2-22-2020002-2000000-2--2--2-2-200--2    complex lifted from SD16
ρ212-2-2220-20002-2000000--2-2-2--2-200--2    complex lifted from SD16
ρ222-2-22-2020002-2000000--2-2-2--2--200-2    complex lifted from SD16
ρ2344-4-4000000004-4000000000000    orthogonal lifted from C8⋊C22
ρ244-4-44000000-4400000000000000    orthogonal lifted from C8⋊C22
ρ254-44-4000000000000002i2i-2i-2i0000    complex lifted from D4.9D4
ρ264-44-400000000000000-2i-2i2i2i0000    complex lifted from D4.9D4

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

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

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

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

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

0100
16000
0010
0001
,
0100
1000
00160
00016
,
51200
5500
0033
00143
,
51200
121200
0033
00314
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,12,5,0,0,0,0,3,14,0,0,3,3],[5,12,0,0,12,12,0,0,0,0,3,3,0,0,3,14] >;

D4.D8 in GAP, Magma, Sage, TeX

D_4.D_8
% in TeX

G:=Group("D4.D8");
// GroupNames label

G:=SmallGroup(128,371);
// by ID

G=gap.SmallGroup(128,371);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

Character table of D4.D8 in TeX

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