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G = C42.289D4  order 128 = 27

271st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.289D4, C42.419C23, C4.612- (1+4), C8⋊Q8⋊15C2, Q8⋊Q8⋊9C2, C4.Q16⋊26C2, C4⋊C8.71C22, (C2×C8).71C23, C4⋊C4.176C23, (C2×C4).435C24, (C22×C4).517D4, C23.300(C2×D4), C4⋊Q8.318C22, C8⋊C4.28C22, C4.Q8.39C22, C22⋊C8.62C22, C42.6C4.3C2, (C2×Q8).167C23, (C4×Q8).114C22, C2.D8.105C22, C4.101(C8.C22), (C2×C42).896C22, Q8⋊C4.49C22, C23.20D4.3C2, C22.695(C22×D4), C22⋊Q8.207C22, C2.66(D8⋊C22), C42.30C22⋊6C2, (C22×C4).1100C23, C42.C2.136C22, C42⋊C2.167C22, C23.37C23.42C2, C2.83(C23.38C23), (C2×C4).559(C2×D4), C2.63(C2×C8.C22), SmallGroup(128,1969)

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

Derived series
C1 — C2×C4 — C42.289D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C23.37C23 — C42.289D4
Lower central
C1 — C2 — C2×C4 — C42.289D4
Upper central
C1 — C22 — C2×C42 — C42.289D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.289D4

Subgroups: 268 in 161 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×12], C22, C22 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×13], Q8 [×10], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×4], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×3], C8⋊C4 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C42⋊C2, C4×Q8 [×2], C4×Q8 [×3], C22⋊Q8 [×2], C22⋊Q8 [×3], C42.C2 [×2], C42.C2, C4⋊Q8 [×4], C42.6C4, Q8⋊Q8 [×2], C4.Q16 [×2], C23.20D4 [×4], C42.30C22 [×2], C8⋊Q8 [×2], C23.37C23 [×2], C42.289D4

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, C42.289D4

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

0161500000
1600150000
00010000
00100000
00004000
0000151300
00000040
0000001513
,
40000000
04000000
00400000
00040000
000013000
00002400
000000130
00000024
,
711500000
106030000
1521070000
2156110000
00000092
000000108
000015900
00007200
,
106030000
711500000
88660000
99770000
00002800
0000101500
00000092
000000108

G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,0,0,1,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,4,15,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,15,0,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,4],[7,10,15,2,0,0,0,0,11,6,2,15,0,0,0,0,5,0,10,6,0,0,0,0,0,3,7,11,0,0,0,0,0,0,0,0,0,0,15,7,0,0,0,0,0,0,9,2,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0],[10,7,8,9,0,0,0,0,6,11,8,9,0,0,0,0,0,5,6,7,0,0,0,0,3,0,6,7,0,0,0,0,0,0,0,0,2,10,0,0,0,0,0,0,8,15,0,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,2,8] >;
 

Character table of C42.289D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ211111-1111-11-1-1-1-1-111-111-1-1-111    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ411111-1111-11-1-1-111-1-11-1-11-1-111    linear of order 2
ρ51111-1-11-11-1-1-1111-11-1-1-111-11-11    linear of order 2
ρ61111-111-111-11-1-1-111-11-11-11-1-11    linear of order 2
ρ71111-1-11-11-1-1-111-11-1111-1-1-11-11    linear of order 2
ρ81111-111-111-11-1-11-1-11-11-111-1-11    linear of order 2
ρ91111-1-11-11-1-1-111-1-11-111-111-11-1    linear of order 2
ρ101111-111-111-11-1-1111-1-11-1-1-111-1    linear of order 2
ρ111111-1-11-11-1-1-11111-11-1-11-11-11-1    linear of order 2
ρ121111-111-111-11-1-1-1-1-111-111-111-1    linear of order 2
ρ1311111111111111-1111-1-1-11-1-1-1-1    linear of order 2
ρ1411111-1111-11-1-1-11-1111-1-1-111-1-1    linear of order 2
ρ15111111111111111-1-1-1111-1-1-1-1-1    linear of order 2
ρ1611111-1111-11-1-1-1-11-1-1-111111-1-1    linear of order 2
ρ172222-22-22-222-2-22000000000000    orthogonal lifted from D4
ρ182222-2-2-22-2-2222-2000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-2-22-22000000000000    orthogonal lifted from D4
ρ20222222-2-2-22-2-22-2000000000000    orthogonal lifted from D4
ρ214-44-40040-400000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2244-4-40-400040000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-44-400-40400000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2444-4-404000-40000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-440004i004i000000000000000    complex lifted from D8⋊C22
ρ264-4-440004i004i000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{289}D_4
 
% in TeX
 
G:=Group("C4^2.289D4");
 
// GroupNames label
 
G:=SmallGroup(128,1969);
 
// by ID
 
G=gap.SmallGroup(128,1969);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,891,100,675,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
 
// generators/relations
 

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