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G = C42.291D4  order 128 = 27

273rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.291D4, C42.421C23, C4.632- (1+4), C8⋊Q8⋊17C2, Q8⋊Q8⋊10C2, D4⋊Q8⋊27C2, C4⋊C8.73C22, (C2×C8).73C23, C4⋊C4.178C23, (C2×C4).437C24, C23.705(C2×D4), (C22×C4).519D4, C4⋊Q8.320C22, C4.127(C8⋊C22), C4.Q8.41C22, C8⋊C4.30C22, C42.6C4⋊19C2, (C2×D4).181C23, (C4×D4).119C22, C22⋊C8.64C22, (C4×Q8).116C22, (C2×Q8).169C23, C2.D8.107C22, D4⋊C4.51C22, C23.48D4⋊23C2, C4⋊D4.204C22, (C2×C42).898C22, Q8⋊C4.51C22, C23.46D4.2C2, C22.697(C22×D4), C22⋊Q8.209C22, C42.28C22⋊8C2, (C22×C4).1102C23, C4.4D4.161C22, C22.22(C8.C22), C42.C2.138C22, C23.36C23.28C2, C2.85(C23.38C23), (C2×C4⋊Q8)⋊45C2, (C2×C4).561(C2×D4), C2.65(C2×C8⋊C22), C2.65(C2×C8.C22), (C2×C4⋊C4).653C22, SmallGroup(128,1971)

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

Derived series
C1 — C2×C4 — C42.291D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C4⋊Q8 — C42.291D4
Lower central
C1 — C2 — C2×C4 — C42.291D4
Upper central
C1 — C22 — C2×C42 — C42.291D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.291D4

Subgroups: 340 in 181 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×19], D4 [×4], Q8 [×10], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×8], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C42.6C4, D4⋊Q8 [×2], Q8⋊Q8 [×2], C23.46D4 [×2], C23.48D4 [×2], C42.28C22 [×2], C8⋊Q8 [×2], C23.36C23, C2×C4⋊Q8, C42.291D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C2×C8⋊C22, C2×C8.C22, C42.291D4

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

00010000
001600000
01000000
160000000
00000107
00001070
000007016
000070160
,
00010000
001600000
01000000
160000000
00000100
00001000
00000001
00000010
,
00330000
001430000
1414000000
314000000
00000010
000000016
000016000
00000100
,
10000000
016000000
001600000
00010000
00001000
000001600
00000010
000000016

G:=sub<GL(8,GF(17))| [0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7,0,0,0,0,1,0,7,0,0,0,0,0,0,7,0,16,0,0,0,0,7,0,16,0],[0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;
 

Character table of C42.291D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11112282222444488888888888
ρ111111111111111111111111111    trivial
ρ21111-1-111111-11-1-1-1111-1-1-1-111-1    linear of order 2
ρ31111-1-11-1-1111-1-111-11-1-1-11-11-11    linear of order 2
ρ41111111-1-111-1-11-1-1-11-111-111-1-1    linear of order 2
ρ5111111-1111111111-1111-1-1-1-1-1-1    linear of order 2
ρ61111-1-1-11111-11-1-1-1-111-1111-1-11    linear of order 2
ρ71111-1-1-1-1-1111-1-11111-1-11-11-11-1    linear of order 2
ρ8111111-1-1-111-1-11-1-111-11-11-1-111    linear of order 2
ρ91111-1-11-1-1111-1-11-1-1-111-111-11-1    linear of order 2
ρ101111111-1-111-1-11-11-1-11-11-1-1-111    linear of order 2
ρ11111111111111111-11-1-1-111-1-1-1-1    linear of order 2
ρ121111-1-111111-11-1-111-1-11-1-11-1-11    linear of order 2
ρ131111-1-1-1-1-1111-1-11-11-1111-1-11-11    linear of order 2
ρ14111111-1-1-111-1-11-111-11-1-1111-1-1    linear of order 2
ρ15111111-111111111-1-1-1-1-1-1-11111    linear of order 2
ρ161111-1-1-11111-11-1-11-1-1-1111-111-1    linear of order 2
ρ172222-2-20-2-2-2-2-222200000000000    orthogonal lifted from D4
ρ182222220-2-2-2-222-2-200000000000    orthogonal lifted from D4
ρ19222222022-2-2-2-2-2200000000000    orthogonal lifted from D4
ρ202222-2-2022-2-22-22-200000000000    orthogonal lifted from D4
ρ2144-4-40004-400000000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4000-4400000000000000000    orthogonal lifted from C8⋊C22
ρ234-44-4000004-4000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ244-4-444-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-44-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-400000-44000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{291}D_4
 
% in TeX
 
G:=Group("C4^2.291D4");
 
// GroupNames label
 
G:=SmallGroup(128,1971);
 
// by ID
 
G=gap.SmallGroup(128,1971);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,436,675,4037,1027,124]);
 
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
 
// generators/relations
 

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