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G = C42.290D4  order 128 = 27

272nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.290D4, C42.420C23, C4.622- (1+4), C8⋊Q8⋊16C2, D4⋊2Q8⋊10C2, C4.Q16⋊27C2, C4⋊C8.72C22, (C2×C8).72C23, C4⋊C4.177C23, (C2×C4).436C24, C23.704(C2×D4), (C22×C4).518D4, C4⋊Q8.319C22, C4.Q8.40C22, C8⋊C4.29C22, C42.6C4⋊18C2, (C4×D4).118C22, (C2×D4).180C23, C22⋊C8.63C22, (C2×Q8).168C23, (C4×Q8).115C22, C22.D8.3C2, C2.D8.106C22, D4⋊C4.50C22, C23.47D4⋊12C2, C4⋊D4.203C22, C4.121(C8.C22), C22.33(C8⋊C22), (C2×C42).897C22, Q8⋊C4.50C22, C22.696(C22×D4), C22⋊Q8.208C22, (C22×C4).1101C23, C42.28C22⋊7C2, C4.4D4.160C22, C42.C2.137C22, C23.36C23.27C2, C2.84(C23.38C23), (C2×C4⋊Q8)⋊44C2, (C2×C4).560(C2×D4), C2.64(C2×C8⋊C22), C2.64(C2×C8.C22), (C2×C4⋊C4).652C22, SmallGroup(128,1970)

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

Derived series
C1 — C2×C4 — C42.290D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C4⋊Q8 — C42.290D4
Lower central
C1 — C2 — C2×C4 — C42.290D4
Upper central
C1 — C22 — C2×C42 — C42.290D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.290D4

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⋊2Q8 [×2], C4.Q16 [×2], C22.D8 [×2], C23.47D4 [×2], C42.28C22 [×2], C8⋊Q8 [×2], C23.36C23, C2×C4⋊Q8, C42.290D4

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, C42.290D4

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=a2b2c3 >

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

48000000
013000000
881380000
00040000
000001116
00001601616
000011601
00001616160
,
1615000000
01000000
0151620000
00010000
00000100
000016000
000000016
00000010
,
101600000
1601160000
201600000
11000000
00000001
00000010
00001000
000001600
,
10000000
1616000000
00100000
1601160000
00001000
000001600
00000001
00000010

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

Character table of C42.290D4

 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
ρ214-4-44-4400000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-444-400000000000000000000    orthogonal lifted from C8⋊C22
ρ234-44-400000-44000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2444-4-4000-4400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-44-4000004-4000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2644-4-40004-400000000000000000    symplectic lifted from C8.C22, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{290}D_4
 
% in TeX
 
G:=Group("C4^2.290D4");
 
// GroupNames label
 
G:=SmallGroup(128,1970);
 
// by ID
 
G=gap.SmallGroup(128,1970);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,100,675,4037,1027,124]);
 
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=a^2*b^2*c^3>;
 
// generators/relations
 

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