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G = D4.7D8order 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⋊C814C2, C4⋊C4.30D4, C82Q81C2, C4.30(C2×D8), D4⋊Q84C2, (C2×D4).53D4, D46D4.2C2, C4.38(C4○D8), (C4×C8).45C22, C4⋊Q8.28C22, D4.D433C2, C4.10D812C2, 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

C1C42 — D4.7D8
C1C2C22C2×C4C42C4×D4D46D4 — D4.7D8
C1C22C42 — D4.7D8
C1C22C42 — D4.7D8
C1C22C22C42 — 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 [×3], C2 [×3], C4 [×4], C4 [×6], C22, C22 [×7], C8 [×4], C2×C4 [×3], C2×C4 [×13], D4 [×2], D4 [×7], Q8 [×4], C23 [×2], C42, C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×3], SD16 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C2.D8 [×3], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22.D4 [×2], C4⋊Q8 [×2], C2×SD16, C2×C4○D4, D4⋊C8 [×2], C4.10D8, D4.D4, D4⋊Q8, C82Q8, D46D4, D4.7D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], 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
ρ152-22-2-220020-20000000-222-20-202    orthogonal lifted from D8
ρ162-22-2-220020-200000002-2-22020-2    orthogonal lifted from D8
ρ172-22-22-20020-200000002-2-220-202    orthogonal lifted from D8
ρ182-22-22-20020-20000000-222-2020-2    orthogonal lifted from D8
ρ1922-2-200020-200-2i2i000022-2-2--20-20    complex lifted from C4○D8
ρ2022-2-200020-200-2i2i0000-2-222-20--20    complex lifted from C4○D8
ρ2122-2-200020-2002i-2i0000-2-222--20-20    complex lifted from C4○D8
ρ2222-2-200020-2002i-2i000022-2-2-20--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 25 54)(2 13 26 55)(3 14 27 56)(4 15 28 49)(5 16 29 50)(6 9 30 51)(7 10 31 52)(8 11 32 53)(17 42 60 38)(18 43 61 39)(19 44 62 40)(20 45 63 33)(21 46 64 34)(22 47 57 35)(23 48 58 36)(24 41 59 37)
(1 18)(2 44)(3 63)(4 34)(5 22)(6 48)(7 59)(8 38)(9 23)(10 41)(11 60)(12 39)(13 19)(14 45)(15 64)(16 35)(17 53)(20 27)(21 49)(24 31)(25 61)(26 40)(28 46)(29 57)(30 36)(32 42)(33 56)(37 52)(43 54)(47 50)(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 48 25 36)(2 47 26 35)(3 46 27 34)(4 45 28 33)(5 44 29 40)(6 43 30 39)(7 42 31 38)(8 41 32 37)(9 18 51 61)(10 17 52 60)(11 24 53 59)(12 23 54 58)(13 22 55 57)(14 21 56 64)(15 20 49 63)(16 19 50 62)

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

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

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

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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