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G = D82D4order 128 = 27

1st semidirect product of D8 and D4 acting via D4/C4=C2

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

Aliases: D82D4, C42D16, C42.136D4, C4⋊C165C2, (C4×D8)⋊2C2, (C2×D16)⋊4C2, C84D46C2, (C2×C8).65D4, C2.5(C2×D16), C8.70(C2×D4), C2.D166C2, (C2×C4).147D8, C8.88(C4○D4), (C4×C8).61C22, (C2×C16).4C22, (C2×D8).2C22, C2.18(C4⋊D8), C4.49(C4⋊D4), C4.14(C8⋊C22), C2.9(C16⋊C22), (C2×C8).514C23, C22.100(C2×D8), C2.D8.156C22, (C2×C4).782(C2×D4), SmallGroup(128,938)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D82D4
C1C2C4C8C2×C8C2×D8C4×D8 — D82D4
C1C2C4C2×C8 — D82D4
C1C22C42C4×C8 — D82D4
C1C2C2C2C2C4C4C2×C8 — D82D4

Generators and relations for D82D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=a5b, dcd=c-1 >

Subgroups: 316 in 93 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], C8 [×2], C8, C2×C4 [×3], C2×C4 [×3], D4 [×11], C23 [×3], C16 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], D8 [×7], C22×C4, C2×D4 [×5], C4×C8, D4⋊C4, C2.D8, C2×C16 [×2], D16 [×4], C4×D4, C41D4, C2×D8, C2×D8 [×2], C2×D8, C2.D16 [×2], C4⋊C16, C4×D8, C84D4, C2×D16 [×2], D82D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, D16 [×2], C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C2×D16, C16⋊C22, D82D4

Character table of D82D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111881616222248822224444444444
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-111111-1-111111111111111    linear of order 2
ρ31111-1-11-11-1-11-1111111-1-1-1111-1-1-11    linear of order 2
ρ4111111-111-1-11-1-1-11111-1-1-1111-1-1-11    linear of order 2
ρ51111-1-11111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ6111111-1-11111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111111-11-1-11-1-1-11111-1-11-1-1-1111-1    linear of order 2
ρ81111-1-1-111-1-11-1111111-1-11-1-1-1111-1    linear of order 2
ρ9222200002-2-22-200-2-2-2-22200000000    orthogonal lifted from D4
ρ102-2-22-2200200-20002-22-20000000000    orthogonal lifted from D4
ρ11222200002222200-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-222-200200-20002-22-20000000000    orthogonal lifted from D4
ρ1322220000-222-2-20000000022-222-2-2-2    orthogonal lifted from D8
ρ1422220000-2-2-2-2200000000-22-22-222-2    orthogonal lifted from D8
ρ1522220000-2-2-2-22000000002-22-22-2-22    orthogonal lifted from D8
ρ1622220000-222-2-200000000-2-22-2-2222    orthogonal lifted from D8
ρ172-22-2000002-2000022-2-22-21671616716165163ζ16716ζ16716165163ζ165163ζ165163    orthogonal lifted from D16
ρ182-22-2000002-20000-2-222-22165163165163ζ16716ζ165163ζ165163ζ167161671616716    orthogonal lifted from D16
ρ192-22-200000-22000022-2-2-22ζ1671616716165163ζ1671616716ζ165163165163ζ165163    orthogonal lifted from D16
ρ202-22-2000002-2000022-2-22-2ζ16716ζ16716ζ1651631671616716ζ165163165163165163    orthogonal lifted from D16
ρ212-22-200000-220000-2-2222-2165163ζ16516316716165163ζ165163ζ1671616716ζ16716    orthogonal lifted from D16
ρ222-22-200000-220000-2-2222-2ζ165163165163ζ16716ζ16516316516316716ζ1671616716    orthogonal lifted from D16
ρ232-22-2000002-20000-2-222-22ζ165163ζ1651631671616516316516316716ζ16716ζ16716    orthogonal lifted from D16
ρ242-22-200000-22000022-2-2-2216716ζ16716ζ16516316716ζ16716165163ζ165163165163    orthogonal lifted from D16
ρ252-2-220000200-20-2i2i-22-220000000000    complex lifted from C4○D4
ρ262-2-220000200-202i-2i-22-220000000000    complex lifted from C4○D4
ρ274-4-440000-400400000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000022-22-22220000000000    orthogonal lifted from C16⋊C22
ρ2944-4-400000000000-222222-220000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of D82D4
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 43 11 35)(2 44 12 36)(3 45 13 37)(4 46 14 38)(5 47 15 39)(6 48 16 40)(7 41 9 33)(8 42 10 34)(17 57 25 49)(18 58 26 50)(19 59 27 51)(20 60 28 52)(21 61 29 53)(22 62 30 54)(23 63 31 55)(24 64 32 56)
(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(33 37)(34 36)(38 40)(41 45)(42 44)(46 48)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)

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

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

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

Matrix representation of D82D4 in GL4(𝔽17) generated by

31400
3300
00160
00016
,
4600
61300
0064
00411
,
16000
01600
00716
001610
,
1000
01600
00160
0031
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[4,6,0,0,6,13,0,0,0,0,6,4,0,0,4,11],[16,0,0,0,0,16,0,0,0,0,7,16,0,0,16,10],[1,0,0,0,0,16,0,0,0,0,16,3,0,0,0,1] >;

D82D4 in GAP, Magma, Sage, TeX

D_8\rtimes_2D_4
% in TeX

G:=Group("D8:2D4");
// GroupNames label

G:=SmallGroup(128,938);
// by ID

G=gap.SmallGroup(128,938);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,64,422,1684,438,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of D82D4 in TeX

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