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G = D4⋊3D8  order 128 = 27

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

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

Aliases: D4⋊3D8, C42.186C23, D4⋊C8⋊8C2, C4⋊D8⋊4C2, C4⋊C4.22D4, C4.26(C2×D8), D4⋊6D4⋊1C2, (C2×D4).45D4, C4.4D8⋊1C2, C4⋊C8.3C22, C4⋊Q8.8C22, C4.53(C4○D8), C4.10D8⋊1C2, (C4×C8).14C22, C4.57(C8⋊C22), (C4×D4).21C22, C2.16(D4⋊D4), C2.12(C22⋊D8), C4⋊1D4.10C22, C2.12(D4.8D4), C22.152C22≀C2, (C2×C4).943(C2×D4), SmallGroup(128,357)

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

Derived series
C1 — C42 — D4⋊3D8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4⋊6D4 — D4⋊3D8
Lower central
C1 — C22 — C42 — D4⋊3D8
Upper central
C1 — C22 — C42 — D4⋊3D8
Jennings
C1 — C22 — C22 — C42 — D4⋊3D8

Generators and relations for D4⋊3D8
 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, 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, D8, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, D4⋊C4, C4⋊C8, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊1D4, C4⋊Q8, C2×D8, C2×C4○D4, D4⋊C8, C4.10D8, C4⋊D8, C4.4D8, D4⋊6D4, D4⋊3D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C4○D8, C8⋊C22, C22⋊D8, D4⋊D4, D4.8D4, D4⋊3D8

Character table of D4⋊3D8

 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
ρ152-22-22-200002-2000000√2-√2-√2√20-√20√2    orthogonal lifted from D8
ρ162-22-2-2200002-2000000√2-√2-√2√20√20-√2    orthogonal lifted from D8
ρ172-22-22-200002-2000000-√2√2√2-√20√20-√2    orthogonal lifted from D8
ρ182-22-2-2200002-2000000-√2√2√2-√20-√20√2    orthogonal lifted from D8
ρ1922-2-200002-2002i-2i0000-√-2-√-2√-2√-2-√20√20    complex lifted from C4○D8
ρ2022-2-200002-200-2i2i0000-√-2-√-2√-2√-2√20-√20    complex lifted from C4○D8
ρ2122-2-200002-200-2i2i0000√-2√-2-√-2-√-2-√20√20    complex lifted from C4○D8
ρ2222-2-200002-2002i-2i0000√-2√-2-√-2-√-2√20-√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 D4⋊3D8
►On 64 points
Generators in S64
(1 56 64 34)(2 35 57 49)(3 50 58 36)(4 37 59 51)(5 52 60 38)(6 39 61 53)(7 54 62 40)(8 33 63 55)(9 42 23 28)(10 29 24 43)(11 44 17 30)(12 31 18 45)(13 46 19 32)(14 25 20 47)(15 48 21 26)(16 27 22 41)

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

Matrix representation of D4⋊3D8 ►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] >;
 

D4⋊3D8 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 D4⋊3D8 in TeX

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