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G = D4.7D8  order 128 = 27

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

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

Aliases: D4.7D8, C42.208C23, D4⋊C8⋊14C2, C4⋊C4.30D4, C8⋊2Q8⋊1C2, C4.30(C2×D8), D4⋊Q8⋊4C2, (C2×D4).53D4, D4⋊6D4.2C2, C4.38(C4○D8), (C4×C8).45C22, C4⋊Q8.28C22, D4.D4⋊33C2, C4.10D8⋊12C2, C4⋊C8.169C22, C4.65(C8⋊C22), (C4×D4).37C22, C2.16(C22⋊D8), C4.38(C8.C22), C22.174C22≀C2, C2.23(D4.7D4), C2.12(D4.10D4), (C2×C4).965(C2×D4), SmallGroup(128,379)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — D4.7D8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4⋊6D4 — D4.7D8
Lower central
C1 — C22 — C42 — D4.7D8
Upper central
C1 — C22 — C42 — D4.7D8
Jennings
C1 — C22 — C22 — C42 — D4.7D8

Generators and relations for D4.7D8
 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=c-1 >

Subgroups: 288 in 120 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×SD16, C2×C4○D4, D4⋊C8, C4.10D8, D4.D4, D4⋊Q8, C8⋊2Q8, D4⋊6D4, D4.7D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C22⋊D8, D4.7D4, D4.10D4, D4.7D8

Character table of D4.7D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111144822224448881644448888
ρ111111111111111111111111111    trivial
ρ21111-1-1-111111-1-1-111-1-1-1-1-11111    linear of order 2
ρ31111-1-111111111-1-1-11-1-1-1-1-11-11    linear of order 2
ρ4111111-111111-1-11-1-1-11111-11-11    linear of order 2
ρ511111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-1-111111-1-1-11111111-1-1-1-1    linear of order 2
ρ71111-1-111111111-1-1-1-111111-11-1    linear of order 2
ρ8111111-111111-1-11-1-11-1-1-1-11-11-1    linear of order 2
ρ922220022-22-2-2-2-2000000000000    orthogonal lifted from D4
ρ102222000-2-2-2-220002-2000000000    orthogonal lifted from D4
ρ11222200-22-22-2-222000000000000    orthogonal lifted from D4
ρ122222220-22-22-200-200000000000    orthogonal lifted from D4
ρ132222-2-20-22-22-200200000000000    orthogonal lifted from D4
ρ142222000-2-2-2-22000-22000000000    orthogonal lifted from D4
ρ152-22-2-220020-20000000-√2√2√2-√20-√20√2    orthogonal lifted from D8
ρ162-22-2-220020-20000000√2-√2-√2√20√20-√2    orthogonal lifted from D8
ρ172-22-22-20020-20000000√2-√2-√2√20-√20√2    orthogonal lifted from D8
ρ182-22-22-20020-20000000-√2√2√2-√20√20-√2    orthogonal lifted from D8
ρ1922-2-200020-200-2i2i0000√2√2-√2-√2-√-20√-20    complex lifted from C4○D8
ρ2022-2-200020-200-2i2i0000-√2-√2√2√2√-20-√-20    complex lifted from C4○D8
ρ2122-2-200020-2002i-2i0000-√2-√2√2√2-√-20√-20    complex lifted from C4○D8
ρ2222-2-200020-2002i-2i0000√2√2-√2-√2√-20-√-20    complex lifted from C4○D8
ρ234-44-40000-404000000000000000    orthogonal lifted from C8⋊C22
ρ244-4-44000000000000002-22-20000    symplectic lifted from D4.10D4, Schur index 2
ρ254-4-4400000000000000-22-220000    symplectic lifted from D4.10D4, Schur index 2
ρ2644-4-4000-4040000000000000000    symplectic lifted from C8.C22, Schur index 2

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

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

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

Matrix representation of D4.7D8 ►in GL4(𝔽17) generated by

1000
0100
00130
00134
,
1000
0100
00138
00134
,
31400
3300
00150
00128
,
141400
14300
00213
001415
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,13,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,13,13,0,0,8,4],[3,3,0,0,14,3,0,0,0,0,15,12,0,0,0,8],[14,14,0,0,14,3,0,0,0,0,2,14,0,0,13,15] >;
 

D4.7D8 in GAP, Magma, Sage, TeX 

D_4._7D_8
 
% in TeX
 
G:=Group("D4.7D8");
 
// GroupNames label
 
G:=SmallGroup(128,379);
 
// by ID
 
G=gap.SmallGroup(128,379);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,456,422,1123,570,521,136,2804,1411,718,172]);
 
// 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=c^-1>;
 
// generators/relations
 

Export

Character table of D4.7D8 in TeX

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