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G = C42.287D4  order 128 = 27

269th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.287D4, C42.417C23, C4.592- (1+4), C8⋊Q8⋊13C2, D4⋊2Q8⋊9C2, D4⋊Q8⋊26C2, C4⋊C8.69C22, (C2×C8).69C23, C4⋊C4.174C23, (C2×C4).433C24, (C22×C4).515D4, C23.299(C2×D4), C4⋊Q8.316C22, C4.105(C8⋊C22), C8⋊C4.26C22, C4.Q8.37C22, C42.6C4⋊17C2, (C2×D4).179C23, (C4×D4).117C22, C22⋊C8.60C22, C2.D8.103C22, D4⋊C4.49C22, C4⋊D4.202C22, C23.19D4⋊28C2, C4⋊1D4.173C22, (C2×C42).894C22, C22.693(C22×D4), C2.64(D8⋊C22), (C22×C4).1098C23, C42.29C22⋊6C2, C42.C2.134C22, C42⋊C2.166C22, C23.37C23⋊22C2, C22.26C24.47C2, C2.81(C23.38C23), (C2×C4).557(C2×D4), C2.63(C2×C8⋊C22), SmallGroup(128,1967)

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

Derived series
C1 — C2×C4 — C42.287D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C23.37C23 — C42.287D4
Lower central
C1 — C2 — C2×C4 — C42.287D4
Upper central
C1 — C22 — C2×C42 — C42.287D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.287D4

Subgroups: 364 in 183 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×10], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×12], Q8 [×6], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×3], C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4, C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C4.4D4, C42.C2 [×2], C4⋊1D4, C4⋊Q8 [×3], C2×C4○D4, C42.6C4, D4⋊Q8 [×2], D4⋊2Q8 [×2], C23.19D4 [×4], C42.29C22 [×2], C8⋊Q8 [×2], C22.26C24, C23.37C23, C42.287D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C2×C8⋊C22, D8⋊C22, C42.287D4

Generators and relations
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1, cbc-1=dbd=a2b, dcd=a2b2c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

001600000
1616120000
10000000
1616010000
000030152
00000322
0000215140
00001515014
,
00100000
1116150000
160000000
110160000
00004000
00000400
00000040
00000004
,
1513060000
1515660000
151515150000
150460000
00002203
000015230
000030152
00000141515
,
1616120000
001600000
016000000
00010000
000030152
00000141515
00002203
000015230

G:=sub<GL(8,GF(17))| [0,16,1,16,0,0,0,0,0,16,0,16,0,0,0,0,16,1,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,3,0,2,15,0,0,0,0,0,3,15,15,0,0,0,0,15,2,14,0,0,0,0,0,2,2,0,14],[0,1,16,1,0,0,0,0,0,1,0,1,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[15,15,15,15,0,0,0,0,13,15,15,0,0,0,0,0,0,6,15,4,0,0,0,0,6,6,15,6,0,0,0,0,0,0,0,0,2,15,3,0,0,0,0,0,2,2,0,14,0,0,0,0,0,3,15,15,0,0,0,0,3,0,2,15],[16,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,1,16,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,2,15,0,0,0,0,0,14,2,2,0,0,0,0,15,15,0,3,0,0,0,0,2,15,3,0] >;
 

Character table of C42.287D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11114882222224448888888888
ρ111111111111111111111111111    trivial
ρ21111-111-11-11-1-1-111-1-11-1-111-11-1    linear of order 2
ρ311111-11-1111-11-1-1-1-1-1-1111-1-111    linear of order 2
ρ41111-1-1111-111-11-1-111-1-1-11-111-1    linear of order 2
ρ511111-1-11111111111-111-11-1-1-1-1    linear of order 2
ρ61111-1-1-1-11-11-1-1-111-111-111-11-11    linear of order 2
ρ7111111-1-1111-11-1-1-1-11-11-1111-1-1    linear of order 2
ρ81111-11-111-111-11-1-11-1-1-1111-1-11    linear of order 2
ρ91111-11-111-111-11-1-1-1-1111-1-111-1    linear of order 2
ρ10111111-1-1111-11-1-1-1111-1-1-1-1-111    linear of order 2
ρ111111-1-1-1-11-11-1-1-11111-111-11-11-1    linear of order 2
ρ1211111-1-1111111111-1-1-1-1-1-11111    linear of order 2
ρ131111-1-1111-111-11-1-1-1111-1-11-1-11    linear of order 2
ρ1411111-11-1111-11-1-1-11-11-11-111-1-1    linear of order 2
ρ151111-111-11-11-1-1-1111-1-11-1-1-11-11    linear of order 2
ρ161111111111111111-11-1-11-1-1-1-1-1    linear of order 2
ρ172222200-2-2-2-2-2-22-220000000000    orthogonal lifted from D4
ρ182222-2002-22-222-2-220000000000    orthogonal lifted from D4
ρ192222-200-2-22-2-2222-20000000000    orthogonal lifted from D4
ρ2022222002-2-2-22-2-22-20000000000    orthogonal lifted from D4
ρ2144-4-40004000-400000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4000-4000400000000000000    orthogonal lifted from C8⋊C22
ρ234-44-40000-404000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ244-44-4000040-4000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ254-4-44000004i004i0000000000000    complex lifted from D8⋊C22
ρ264-4-44000004i004i0000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{287}D_4
 
% in TeX
 
G:=Group("C4^2.287D4");
 
// GroupNames label
 
G:=SmallGroup(128,1967);
 
// by ID
 
G=gap.SmallGroup(128,1967);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,891,100,675,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*b^2*c^3>;
 
// generators/relations
 

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