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G = C42.286D4  order 128 = 27

268th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.286D4, C42.416C23, C4.582- (1+4), C8⋊Q8⋊12C2, D4.Q8⋊23C2, C4⋊C8.68C22, (C2×C8).68C23, C4⋊C4.173C23, (C2×C4).432C24, C23.702(C2×D4), (C22×C4).514D4, C4⋊Q8.315C22, C8⋊C4.25C22, C4.Q8.36C22, C42.6C4⋊16C2, (C2×D4).178C23, (C4×D4).116C22, C22.D8⋊23C2, C22⋊C8.59C22, C2.D8.102C22, D4⋊C4.48C22, C4⋊1D4.172C22, C4⋊D4.201C22, C23.46D4⋊12C2, C22.52(C8⋊C22), (C2×C42).893C22, C22.692(C22×D4), C2.63(D8⋊C22), (C22×C4).1097C23, C42.29C22⋊5C2, C42.C2.133C22, C22.26C24.46C2, C2.80(C23.38C23), (C2×C4).556(C2×D4), C2.62(C2×C8⋊C22), (C2×C42.C2)⋊37C2, (C2×C4⋊C4).650C22, SmallGroup(128,1966)

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

Derived series
C1 — C2×C4 — C42.286D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C42.C2 — C42.286D4
Lower central
C1 — C2 — C2×C4 — C42.286D4
Upper central
C1 — C22 — C2×C42 — C42.286D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.286D4

Subgroups: 364 in 184 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×19], D4 [×12], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C42.C2 [×4], C42.C2 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4, C42.6C4, D4.Q8 [×4], C22.D8 [×2], C23.46D4 [×2], C42.29C22 [×2], C8⋊Q8 [×2], C2×C42.C2, C22.26C24, C42.286D4

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, C42.286D4

Generators and relations
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=a2c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

21613130000
520130000
1216360000
5012100000
00001000
00000100
000000160
000000016
,
016000000
10000000
1116150000
016110000
000013000
000001300
00000040
00000004
,
1116150000
00100000
10000000
010160000
00000001
00000010
000016000
00000100
,
10000000
016000000
1116150000
016010000
00001000
000001600
00000001
00000010

G:=sub<GL(8,GF(17))| [2,5,12,5,0,0,0,0,16,2,16,0,0,0,0,0,13,0,3,12,0,0,0,0,13,13,6,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,1,0,0,0,0,0,16,0,1,16,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,16,1,0,0,0,0,0,0,15,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,1,0,0,0,0,0,0,16,1,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

Character table of C42.286D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11112288222244448888888888
ρ111111111111111111111111111    trivial
ρ2111111-11-1-111-1-11-1-11-11-1111-1-1    linear of order 2
ρ31111-1-111-1-1111-1-11-1-11-1-111-11-1    linear of order 2
ρ41111-1-1-111111-11-1-11-1-1-1111-1-11    linear of order 2
ρ51111-1-1-1-1-1-1111-1-11-111-111-11-11    linear of order 2
ρ61111-1-11-11111-11-1-111-1-1-11-111-1    linear of order 2
ρ7111111-1-1111111111-111-11-1-1-1-1    linear of order 2
ρ81111111-1-1-111-1-11-1-1-1-1111-1-111    linear of order 2
ρ91111111-1-1-111-1-11-11-11-11-111-1-1    linear of order 2
ρ10111111-1-111111111-1-1-1-1-1-11111    linear of order 2
ρ111111-1-11-11111-11-1-1-1111-1-11-1-11    linear of order 2
ρ121111-1-1-1-1-1-1111-1-1111-111-11-11-1    linear of order 2
ρ131111-1-1-111111-11-1-1-1-1111-1-111-1    linear of order 2
ρ141111-1-111-1-1111-1-111-1-11-1-1-11-11    linear of order 2
ρ15111111-11-1-111-1-11-1111-1-1-1-1-111    linear of order 2
ρ161111111111111111-11-1-11-1-1-1-1-1    linear of order 2
ρ172222-2-200-2-2-2-2-22220000000000    orthogonal lifted from D4
ρ182222220022-2-2-2-2-220000000000    orthogonal lifted from D4
ρ1922222200-2-2-2-222-2-20000000000    orthogonal lifted from D4
ρ202222-2-20022-2-22-22-20000000000    orthogonal lifted from D4
ρ214-4-444-400000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44-4400000000000000000000    orthogonal lifted from C8⋊C22
ρ234-44-40000004-400000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ244-44-4000000-4400000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2544-4-400004i4i0000000000000000    complex lifted from D8⋊C22
ρ2644-4-400004i4i0000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{286}D_4
 
% in TeX
 
G:=Group("C4^2.286D4");
 
// GroupNames label
 
G:=SmallGroup(128,1966);
 
// by ID
 
G=gap.SmallGroup(128,1966);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,891,100,675,4037,1027,124]);
 
// 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=a^2*b^-1,d*b*d=a^2*b,d*c*d=a^2*c^3>;
 
// generators/relations
 

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