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G = D8.10D4order 128 = 27

2nd non-split extension by D8 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D8.10D4, Q16.9D4, C23.15D8, (C2×Q32)⋊2C2, C8.65(C2×D4), C2.D165C2, C22⋊C168C2, (C2×C8).173D4, (C2×C4).113D8, C2.Q329C2, C4.21C22≀C2, (C2×SD32)⋊10C2, C2.7(C4○D16), C22.99(C2×D8), C8.18D416C2, C4.13(C8⋊C22), C2.D8.4C22, (C2×C8).513C23, (C2×C16).37C22, C2.8(Q32⋊C2), (C22×C4).348D4, C2.29(C22⋊D8), (C2×D8).109C22, (C22×C8).173C22, (C2×Q16).108C22, (C2×C4○D8).5C2, (C2×C4).781(C2×D4), SmallGroup(128,921)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8.10D4
C1C2C4C2×C4C2×C8C22×C8C2×C4○D8 — D8.10D4
C1C2C4C2×C8 — D8.10D4
C1C22C22×C4C22×C8 — D8.10D4
C1C2C2C2C2C4C4C2×C8 — D8.10D4

Generators and relations for D8.10D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a4, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 276 in 110 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C8 [×2], C8, C2×C4 [×2], C2×C4 [×9], D4 [×7], Q8 [×5], C23, C23, C16 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16 [×3], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×6], Q8⋊C4, C2.D8, C2×C16 [×2], SD32 [×2], Q32 [×2], C22⋊Q8, C22×C8, C2×D8, C2×SD16, C2×Q16 [×2], C4○D8 [×4], C2×C4○D4, C22⋊C16, C2.D16, C2.Q32, C8.18D4, C2×SD32, C2×Q32, C2×C4○D8, D8.10D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C4○D16, Q32⋊C2, D8.10D4

Character table of D8.10D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111488222288161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1111-1-11-1-1-111111-1-1-1111-1-1-11    linear of order 2
ρ41111-1111-1-11-1-11-11111-1-11-1-1-1111-1    linear of order 2
ρ511111-1-11111-1-1-1-111111111111111    linear of order 2
ρ611111-1-11111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-11-1-11111-11111-1-1-1111-1-1-11    linear of order 2
ρ81111-1-1-11-1-1111-111111-1-11-1-1-1111-1    linear of order 2
ρ92222-2002-2-220000-2-2-2-22200000000    orthogonal lifted from D4
ρ102-2-22000200-22-2002-22-20000000000    orthogonal lifted from D4
ρ11222220022220000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-220-22200-20000-22-220000000000    orthogonal lifted from D4
ρ132-2-22000200-2-22002-22-20000000000    orthogonal lifted from D4
ρ142-2-2202-2200-20000-22-220000000000    orthogonal lifted from D4
ρ152222200-2-2-2-20000000000-22-22-222-2    orthogonal lifted from D8
ρ162222200-2-2-2-200000000002-22-22-2-22    orthogonal lifted from D8
ρ172222-200-222-20000000000-2-22-2-2222    orthogonal lifted from D8
ρ182222-200-222-2000000000022-222-2-2-2    orthogonal lifted from D8
ρ192-22-200002i-2i00000-222-2-2--2ζ165163ζ16716ζ16516316716ζ16131611ζ16716ζ1615169165163    complex lifted from C4○D16
ρ202-22-20000-2i2i00000-222-2--2-2ζ16131611ζ16716ζ16516316716ζ165163ζ1615169ζ16716165163    complex lifted from C4○D16
ρ212-22-200002i-2i00000-222-2-2--2ζ1613161116716165163ζ16716ζ165163ζ1615169ζ16716ζ165163    complex lifted from C4○D16
ρ222-22-20000-2i2i00000-222-2--2-2ζ16516316716165163ζ16716ζ16131611ζ16716ζ1615169ζ165163    complex lifted from C4○D16
ρ232-22-20000-2i2i000002-2-22-2--2ζ16716165163ζ16716ζ165163ζ1615169ζ16131611ζ16516316716    complex lifted from C4○D16
ρ242-22-200002i-2i000002-2-22--2-2ζ1615169165163ζ16716ζ165163ζ16716ζ165163ζ1613161116716    complex lifted from C4○D16
ρ252-22-20000-2i2i000002-2-22-2--2ζ1615169ζ16516316716165163ζ16716ζ165163ζ16131611ζ16716    complex lifted from C4○D16
ρ262-22-200002i-2i000002-2-22--2-2ζ16716ζ16516316716165163ζ1615169ζ16131611ζ165163ζ16716    complex lifted from C4○D16
ρ274-4-44000-4004000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2944-4-4000000000002222-22-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of D8.10D4
On 64 points
Generators in S64
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 40)(34 39)(35 38)(36 37)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(57 61)(58 60)(62 64)
(1 16 19 64)(2 15 20 63)(3 14 21 62)(4 13 22 61)(5 12 23 60)(6 11 24 59)(7 10 17 58)(8 9 18 57)(25 52 36 41)(26 51 37 48)(27 50 38 47)(28 49 39 46)(29 56 40 45)(30 55 33 44)(31 54 34 43)(32 53 35 42)
(1 44 5 48)(2 43 6 47)(3 42 7 46)(4 41 8 45)(9 40 13 36)(10 39 14 35)(11 38 15 34)(12 37 16 33)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 57 29 61)(26 64 30 60)(27 63 31 59)(28 62 32 58)

G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,40)(34,39)(35,38)(36,37)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64), (1,16,19,64)(2,15,20,63)(3,14,21,62)(4,13,22,61)(5,12,23,60)(6,11,24,59)(7,10,17,58)(8,9,18,57)(25,52,36,41)(26,51,37,48)(27,50,38,47)(28,49,39,46)(29,56,40,45)(30,55,33,44)(31,54,34,43)(32,53,35,42), (1,44,5,48)(2,43,6,47)(3,42,7,46)(4,41,8,45)(9,40,13,36)(10,39,14,35)(11,38,15,34)(12,37,16,33)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,57,29,61)(26,64,30,60)(27,63,31,59)(28,62,32,58)>;

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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,40)(34,39)(35,38)(36,37)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64), (1,16,19,64)(2,15,20,63)(3,14,21,62)(4,13,22,61)(5,12,23,60)(6,11,24,59)(7,10,17,58)(8,9,18,57)(25,52,36,41)(26,51,37,48)(27,50,38,47)(28,49,39,46)(29,56,40,45)(30,55,33,44)(31,54,34,43)(32,53,35,42), (1,44,5,48)(2,43,6,47)(3,42,7,46)(4,41,8,45)(9,40,13,36)(10,39,14,35)(11,38,15,34)(12,37,16,33)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,57,29,61)(26,64,30,60)(27,63,31,59)(28,62,32,58) );

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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,8),(2,7),(3,6),(4,5),(9,13),(10,12),(14,16),(17,20),(18,19),(21,24),(22,23),(25,26),(27,32),(28,31),(29,30),(33,40),(34,39),(35,38),(36,37),(41,45),(42,44),(46,48),(49,51),(52,56),(53,55),(57,61),(58,60),(62,64)], [(1,16,19,64),(2,15,20,63),(3,14,21,62),(4,13,22,61),(5,12,23,60),(6,11,24,59),(7,10,17,58),(8,9,18,57),(25,52,36,41),(26,51,37,48),(27,50,38,47),(28,49,39,46),(29,56,40,45),(30,55,33,44),(31,54,34,43),(32,53,35,42)], [(1,44,5,48),(2,43,6,47),(3,42,7,46),(4,41,8,45),(9,40,13,36),(10,39,14,35),(11,38,15,34),(12,37,16,33),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(25,57,29,61),(26,64,30,60),(27,63,31,59),(28,62,32,58)])

Matrix representation of D8.10D4 in GL4(𝔽17) generated by

16000
01600
00011
00311
,
16000
0100
00011
00140
,
0100
16000
00105
0047
,
0100
1000
001114
0016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,1,0,0,0,0,0,14,0,0,11,0],[0,16,0,0,1,0,0,0,0,0,10,4,0,0,5,7],[0,1,0,0,1,0,0,0,0,0,11,1,0,0,14,6] >;

D8.10D4 in GAP, Magma, Sage, TeX

D_8._{10}D_4
% in TeX

G:=Group("D8.10D4");
// GroupNames label

G:=SmallGroup(128,921);
// by ID

G=gap.SmallGroup(128,921);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,352,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D8.10D4 in TeX

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