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G = C42.277D4  order 128 = 27

259th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.277D4, C42.405C23, C4.1112+ (1+4), C8⋊3D4⋊12C2, D4⋊D4⋊27C2, C8.2D4⋊12C2, C4⋊C8.64C22, (C2×C8).67C23, D4.2D4⋊24C2, D4.7D4⋊28C2, C4⋊C4.158C23, (C2×C4).417C24, Q8.D4⋊24C2, (C2×D8).72C22, (C22×C4).506D4, C23.289(C2×D4), C4⋊Q8.307C22, C8⋊C4.21C22, C42.6C4⋊15C2, (C4×D4).108C22, (C2×D4).166C23, C22⋊C8.52C22, (C2×Q8).154C23, (C4×Q8).105C22, (C2×Q16).72C22, D4⋊C4.46C22, C4⋊1D4.168C22, C4⋊D4.195C22, (C2×C42).884C22, Q8⋊C4.47C22, (C2×SD16).36C22, C22.677(C22×D4), C22⋊Q8.200C22, C2.60(D8⋊C22), C42.29C22⋊2C2, C22.26C24⋊21C2, (C22×C4).1088C23, C42.30C22⋊2C2, C4.4D4.157C22, C42.C2.128C22, C23.36C23⋊12C2, C2.88(C22.29C24), (C2×C4).546(C2×D4), (C2×C4○D4).176C22, SmallGroup(128,1951)

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

Derived series
C1 — C2×C4 — C42.277D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.26C24 — C42.277D4
Lower central
C1 — C2 — C2×C4 — C42.277D4
Upper central
C1 — C22 — C2×C42 — C42.277D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.277D4

Subgroups: 420 in 198 conjugacy classes, 84 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×16], Q8 [×6], C23, C23 [×3], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4, C4.4D4, C4.4D4 [×2], C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C2×C4○D4 [×2], C42.6C4, D4⋊D4 [×2], D4.7D4 [×2], D4.2D4 [×2], Q8.D4 [×2], C42.29C22, C42.30C22, C8⋊3D4, C8.2D4, C23.36C23, C22.26C24, C42.277D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×2], C22.29C24, D8⋊C22 [×2], C42.277D4

Generators and relations
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=b2c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

00100000
00010000
160000000
016000000
000000130
000000013
00004000
00000400
,
01000000
10000000
00010000
00100000
00000010
00000001
000016000
000001600
,
00350000
0012140000
1412000000
53000000
00007722
000050160
0000221010
0000160120
,
00350000
0012140000
35000000
1214000000
00007722
00005101615
0000221010
00001615127

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,14,5,0,0,0,0,0,0,12,3,0,0,0,0,3,12,0,0,0,0,0,0,5,14,0,0,0,0,0,0,0,0,0,0,7,5,2,16,0,0,0,0,7,0,2,0,0,0,0,0,2,16,10,12,0,0,0,0,2,0,10,0],[0,0,3,12,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,5,14,0,0,0,0,0,0,0,0,0,0,7,5,2,16,0,0,0,0,7,10,2,15,0,0,0,0,2,16,10,12,0,0,0,0,2,15,10,7] >;
 

Character table of C42.277D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11114888222222444888888888
ρ111111111111111111111111111    trivial
ρ211111-1-1-1111111111-1-1-1-1-11111    linear of order 2
ρ3111111-111111111111-1-11-1-1-1-1-1    linear of order 2
ρ411111-11-1111111111-111-11-1-1-1-1    linear of order 2
ρ51111-11-11-1-11-11-11-11-1-11-11-11-11    linear of order 2
ρ61111-1-11-1-1-11-11-11-1111-11-1-11-11    linear of order 2
ρ71111-1111-1-11-11-11-11-11-1-1-11-11-1    linear of order 2
ρ81111-1-1-1-1-1-11-11-11-111-11111-11-1    linear of order 2
ρ9111111-1-1-111-111-1-1-1-1111-1-1-111    linear of order 2
ρ1011111-111-111-111-1-1-11-1-1-11-1-111    linear of order 2
ρ111111111-1-111-111-1-1-1-1-1-11111-1-1    linear of order 2
ρ1211111-1-11-111-111-1-1-1111-1-111-1-1    linear of order 2
ρ131111-111-11-1111-1-11-11-11-1-11-1-11    linear of order 2
ρ141111-1-1-111-1111-1-11-1-11-1111-1-11    linear of order 2
ρ151111-11-1-11-1111-1-11-111-1-11-111-1    linear of order 2
ρ161111-1-1111-1111-1-11-1-1-111-1-111-1    linear of order 2
ρ172222-200022-22-22-2-22000000000    orthogonal lifted from D4
ρ182222-2000-22-2-2-2222-2000000000    orthogonal lifted from D4
ρ19222220002-2-22-2-22-2-2000000000    orthogonal lifted from D4
ρ2022222000-2-2-2-2-2-2-222000000000    orthogonal lifted from D4
ρ214-44-4000000-4040000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-400000040-40000000000000    orthogonal lifted from 2+ (1+4)
ρ234-4-44000004i0004i000000000000    complex lifted from D8⋊C22
ρ2444-4-400004i004i00000000000000    complex lifted from D8⋊C22
ρ2544-4-400004i004i00000000000000    complex lifted from D8⋊C22
ρ264-4-44000004i0004i000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{277}D_4
 
% in TeX
 
G:=Group("C4^2.277D4");
 
// GroupNames label
 
G:=SmallGroup(128,1951);
 
// by ID
 
G=gap.SmallGroup(128,1951);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,675,248,4037,1027,124]);
 
// 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*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=b^2*c^3>;
 
// generators/relations
 

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