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G = D43D8order 128 = 27

2nd semidirect product of D4 and D8 acting via D8/D4=C2

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

Aliases: D43D8, C42.186C23, D4⋊C88C2, C4⋊D84C2, C4⋊C4.22D4, C4.26(C2×D8), D46D41C2, (C2×D4).45D4, C4.4D81C2, C4⋊C8.3C22, C4⋊Q8.8C22, C4.53(C4○D8), C4.10D81C2, (C4×C8).14C22, C4.57(C8⋊C22), (C4×D4).21C22, C2.16(D4⋊D4), C2.12(C22⋊D8), C41D4.10C22, C2.12(D4.8D4), C22.152C22≀C2, (C2×C4).943(C2×D4), SmallGroup(128,357)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — D43D8
C1C2C22C2×C4C42C4×D4D46D4 — D43D8
C1C22C42 — D43D8
C1C22C42 — D43D8
C1C22C22C42 — D43D8

Generators and relations for D43D8
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 344 in 128 conjugacy classes, 36 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×5], C22, C22 [×10], C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×2], D4 [×13], Q8 [×2], C23 [×3], C42, C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×3], D8 [×4], C22×C4 [×4], C2×D4 [×2], C2×D4 [×5], C2×Q8, C4○D4 [×4], C4×C8, D4⋊C4 [×4], C4⋊C8 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22.D4 [×2], C41D4, C4⋊Q8, C2×D8 [×2], C2×C4○D4, D4⋊C8 [×2], C4.10D8, C4⋊D8 [×2], C4.4D8, D46D4, D43D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C4○D8, C8⋊C22 [×2], C22⋊D8, D4⋊D4, D4.8D4, D43D8

Character table of D43D8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 111144816222244488844448888
ρ111111111111111111111111111    trivial
ρ21111-1-1111111111-1-1-1-1-1-1-1-11-11    linear of order 2
ρ31111-1-1-111111-1-1111-11111-1-1-1-1    linear of order 2
ρ4111111-111111-1-11-1-11-1-1-1-11-11-1    linear of order 2
ρ51111-1-11-11111111-1-1-111111-11-1    linear of order 2
ρ61111111-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-1-11111-1-11-1-111111-11-11    linear of order 2
ρ81111-1-1-1-11111-1-1111-1-1-1-1-11111    linear of order 2
ρ922220000-2-2-2-20022-2000000000    orthogonal lifted from D4
ρ10222200-2022-2-222-200000000000    orthogonal lifted from D4
ρ1122222200-2-22200-200-200000000    orthogonal lifted from D4
ρ122222002022-2-2-2-2-200000000000    orthogonal lifted from D4
ρ1322220000-2-2-2-2002-22000000000    orthogonal lifted from D4
ρ142222-2-200-2-22200-200200000000    orthogonal lifted from D4
ρ152-22-22-200002-20000002-2-220-202    orthogonal lifted from D8
ρ162-22-2-2200002-20000002-2-22020-2    orthogonal lifted from D8
ρ172-22-22-200002-2000000-222-2020-2    orthogonal lifted from D8
ρ182-22-2-2200002-2000000-222-20-202    orthogonal lifted from D8
ρ1922-2-200002-2002i-2i0000--2--2-2-2-2020    complex lifted from C4○D8
ρ2022-2-200002-200-2i2i0000--2--2-2-220-20    complex lifted from C4○D8
ρ2122-2-200002-200-2i2i0000-2-2--2--2-2020    complex lifted from C4○D8
ρ2222-2-200002-2002i-2i0000-2-2--2--220-20    complex lifted from C4○D8
ρ234-44-4000000-4400000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-40000-440000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-44000000000000002i-2i2i-2i0000    complex lifted from D4.8D4
ρ264-4-4400000000000000-2i2i-2i2i0000    complex lifted from D4.8D4

Smallest permutation representation of D43D8
On 64 points
Generators in S64
(1 56 15 28)(2 29 16 49)(3 50 9 30)(4 31 10 51)(5 52 11 32)(6 25 12 53)(7 54 13 26)(8 27 14 55)(17 36 58 44)(18 45 59 37)(19 38 60 46)(20 47 61 39)(21 40 62 48)(22 41 63 33)(23 34 64 42)(24 43 57 35)
(1 43)(2 58)(3 45)(4 60)(5 47)(6 62)(7 41)(8 64)(9 37)(10 19)(11 39)(12 21)(13 33)(14 23)(15 35)(16 17)(18 50)(20 52)(22 54)(24 56)(25 40)(26 63)(27 34)(28 57)(29 36)(30 59)(31 38)(32 61)(42 55)(44 49)(46 51)(48 53)
(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)
(2 8)(3 7)(4 6)(9 13)(10 12)(14 16)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 51)(26 50)(27 49)(28 56)(29 55)(30 54)(31 53)(32 52)(33 59)(34 58)(35 57)(36 64)(37 63)(38 62)(39 61)(40 60)

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

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

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

Matrix representation of D43D8 in GL4(𝔽17) generated by

161500
1100
0010
0001
,
6600
141100
00160
00016
,
4800
01300
0006
00146
,
1000
161600
0010
00116
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[6,14,0,0,6,11,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,8,13,0,0,0,0,0,14,0,0,6,6],[1,16,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

D43D8 in GAP, Magma, Sage, TeX

D_4\rtimes_3D_8
% in TeX

G:=Group("D4:3D8");
// GroupNames label

G:=SmallGroup(128,357);
// by ID

G=gap.SmallGroup(128,357);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

Character table of D43D8 in TeX

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