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G = D8.5D4order 128 = 27

2nd non-split extension by D8 of D4 acting via D4/C4=C2

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

Aliases: D8.5D4, C42.140D4, C4⋊C167C2, (C4×D8)⋊19C2, (C2×C4).34D8, C8.74(C2×D4), (C2×D16).2C2, (C2×C8).174D4, C2.D1612C2, C2.Q326C2, (C2×SD32)⋊14C2, C8.92(C4○D4), C2.8(C4○D16), C8.12D41C2, C4.53(C4⋊D4), C2.22(C4⋊D8), C4.18(C8⋊C22), (C2×C16).41C22, (C2×C8).518C23, (C4×C8).103C22, C22.104(C2×D8), (C2×Q16).4C22, C2.11(C16⋊C22), (C2×D8).111C22, C2.D8.160C22, (C2×C4).786(C2×D4), SmallGroup(128,942)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8.5D4
C1C2C4C8C2×C8C2×D8C4×D8 — D8.5D4
C1C2C4C2×C8 — D8.5D4
C1C22C42C4×C8 — D8.5D4
C1C2C2C2C2C4C4C2×C8 — D8.5D4

Generators and relations for D8.5D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=a4c-1 >

Subgroups: 252 in 85 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], D4 [×5], Q8 [×2], C23 [×2], C16 [×2], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], D8 [×3], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C2×Q8, C4×C8, D4⋊C4, C2.D8, C2×C16 [×2], D16 [×2], SD32 [×2], C4×D4, C4.4D4, C2×D8 [×2], C2×SD16, C2×Q16, C2.D16, C2.Q32, C4⋊C16, C4×D8, C8.12D4, C2×D16, C2×SD32, D8.5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C4○D16, C16⋊C22, D8.5D4

Character table of D8.5D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111881622224881622224444444444
ρ111111111111111111111111111111    trivial
ρ2111111-11111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-11-1-11-11111111-1-11-1-11-111-1    linear of order 2
ρ41111-1-111-1-11-111-11111-1-1-111-11-1-11    linear of order 2
ρ5111111-11-1-11-1-1-111111-1-1-111-11-1-11    linear of order 2
ρ611111111-1-11-1-1-1-11111-1-11-1-11-111-1    linear of order 2
ρ71111-1-1111111-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-111111-1-1-111111111111111    linear of order 2
ρ92-2-222-20200-20000-222-20000000000    orthogonal lifted from D4
ρ1022220002-2-22-2000-2-2-2-22200000000    orthogonal lifted from D4
ρ11222200022222000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-22-220200-20000-222-20000000000    orthogonal lifted from D4
ρ132222000-222-2-2000000000-2-2-222-222    orthogonal lifted from D8
ρ142222000-222-2-2000000000222-2-22-2-2    orthogonal lifted from D8
ρ152222000-2-2-2-22000000000-2222-2-22-2    orthogonal lifted from D8
ρ162222000-2-2-2-220000000002-2-2-222-22    orthogonal lifted from D8
ρ172-2-22000200-202i-2i02-2-220000000000    complex lifted from C4○D4
ρ182-2-22000200-20-2i2i02-2-220000000000    complex lifted from C4○D4
ρ192-22-20000-2i2i00000-22-22--2-2ζ16716ζ16131611ζ165163165163ζ161516916716ζ165163ζ16716    complex lifted from C4○D16
ρ202-22-200002i-2i00000-22-22-2--2ζ16716ζ165163ζ16131611165163ζ1671616716ζ165163ζ1615169    complex lifted from C4○D16
ρ212-22-200002i-2i000002-22-2--2-2ζ165163ζ16716ζ1615169ζ16716ζ1613161116516316716ζ165163    complex lifted from C4○D16
ρ222-22-20000-2i2i000002-22-2-2--2165163ζ16716ζ161516916716ζ16131611ζ165163ζ16716ζ165163    complex lifted from C4○D16
ρ232-22-20000-2i2i000002-22-2-2--2ζ165163ζ1615169ζ16716ζ16716ζ16516316516316716ζ16131611    complex lifted from C4○D16
ρ242-22-200002i-2i000002-22-2--2-2165163ζ1615169ζ1671616716ζ165163ζ165163ζ16716ζ16131611    complex lifted from C4○D16
ρ252-22-200002i-2i00000-22-22-2--216716ζ16131611ζ165163ζ165163ζ1615169ζ16716165163ζ16716    complex lifted from C4○D16
ρ262-22-20000-2i2i00000-22-22--2-216716ζ165163ζ16131611ζ165163ζ16716ζ16716165163ζ1615169    complex lifted from C4○D16
ρ274-4-44000-4004000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000-22-2222220000000000    orthogonal lifted from C16⋊C22
ρ2944-4-4000000000002222-22-220000000000    orthogonal lifted from C16⋊C22

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

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

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

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

Matrix representation of D8.5D4 in GL4(𝔽17) generated by

31400
3300
0010
0001
,
131100
11400
0010
0001
,
4000
0400
00115
00116
,
4000
01300
00115
00016
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,1,0,0,0,0,1],[13,11,0,0,11,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,15,16] >;

D8.5D4 in GAP, Magma, Sage, TeX

D_8._5D_4
% in TeX

G:=Group("D8.5D4");
// GroupNames label

G:=SmallGroup(128,942);
// by ID

G=gap.SmallGroup(128,942);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,1684,438,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of D8.5D4 in TeX

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