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G = C42.301D4  order 128 = 27

283rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.301D4, C4.92- (1+4), C42.435C23, C4.282+ (1+4), C4⋊D8⋊30C2, C8⋊2D4⋊16C2, D4⋊2Q8⋊14C2, C4⋊C8.81C22, (C2×C8).77C23, C4⋊C4.192C23, (C2×C4).451C24, (C2×D8).75C22, C23.308(C2×D4), (C22×C4).528D4, C4⋊Q8.329C22, C4.106(C8⋊C22), C4⋊M4(2)⋊12C2, C4.Q8.47C22, (C2×D4).193C23, (C4×D4).131C22, D4⋊C4.58C22, C4⋊1D4.178C22, C4⋊D4.213C22, (C2×C42).908C22, C22.711(C22×D4), (C22×C4).1106C23, C22.26C24⋊25C2, (C2×M4(2)).89C22, C2.70(C22.31C24), (C2×C4).575(C2×D4), C2.68(C2×C8⋊C22), SmallGroup(128,1985)

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

Derived series
C1 — C2×C4 — C42.301D4
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.26C24 — C42.301D4
Lower central
C1 — C2 — C2×C4 — C42.301D4
Upper central
C1 — C22 — C2×C42 — C42.301D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.301D4

Subgroups: 484 in 215 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×24], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×2], D8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C4○D4 [×8], D4⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2×C42, C4×D4 [×4], C4×D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×D8 [×4], C2×C4○D4 [×2], C4⋊M4(2), C4⋊D8 [×4], C8⋊2D4 [×4], D4⋊2Q8 [×4], C22.26C24 [×2], C42.301D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×4], C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, C2×C8⋊C22 [×2], C42.301D4

Generators and relations
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=ab2, cbc-1=dbd=b-1, dcd=a2c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

160100000
001610000
00100000
01100000
00000010
0000111615
000016000
000011016
,
1615000000
11000000
0151150000
16151160000
00000010
0000111615
000016000
000011016
,
130000000
44000000
001300000
1301340000
00009977
0000816010
00008108
00009889
,
1615000000
01000000
15151620000
1515010000
00000010
0000161612
00001000
000016011

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

Character table of C42.301D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11114888822222244488888888
ρ111111111111111111111111111    trivial
ρ21111-1-11-1-11-1111-1-11-11-1111-1-11    linear of order 2
ρ31111111-1-1-1-11-11-1-1-1111-1-1-11-11    linear of order 2
ρ41111-1-1111-111-1111-1-11-1-1-1-1-111    linear of order 2
ρ511111-1-1-1-1111111111-1-1-1-11111    linear of order 2
ρ61111-11-1111-1111-1-11-1-11-1-11-1-11    linear of order 2
ρ711111-1-111-1-11-11-1-1-11-1-111-11-11    linear of order 2
ρ81111-11-1-1-1-111-1111-1-1-1111-1-111    linear of order 2
ρ9111111-11-1-1-11-11-1-1-111-11-11-11-1    linear of order 2
ρ101111-1-1-1-11-111-1111-1-1111-111-1-1    linear of order 2
ρ11111111-1-111111111111-1-11-1-1-1-1    linear of order 2
ρ121111-1-1-11-11-1111-1-11-111-11-111-1    linear of order 2
ρ1311111-11-11-1-11-11-1-1-11-11-111-11-1    linear of order 2
ρ141111-1111-1-111-1111-1-1-1-1-1111-1-1    linear of order 2
ρ1511111-111-1111111111-111-1-1-1-1-1    linear of order 2
ρ161111-111-111-1111-1-11-1-1-11-1-111-1    linear of order 2
ρ172222-2000022-22-22-2-2200000000    orthogonal lifted from D4
ρ182222-20000-2-2-2-2-2-222200000000    orthogonal lifted from D4
ρ192222200002-2-22-2-22-2-200000000    orthogonal lifted from D4
ρ20222220000-22-2-2-22-22-200000000    orthogonal lifted from D4
ρ214-44-40000000-404000000000000    orthogonal lifted from 2+ (1+4)
ρ224-4-44000000-4000400000000000    orthogonal lifted from C8⋊C22
ρ2344-4-400000-40040000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400000400-40000000000000    orthogonal lifted from C8⋊C22
ρ254-4-440000004000-400000000000    orthogonal lifted from C8⋊C22
ρ264-44-4000000040-4000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{301}D_4
 
% in TeX
 
G:=Group("C4^2.301D4");
 
// GroupNames label
 
G:=SmallGroup(128,1985);
 
// by ID
 
G=gap.SmallGroup(128,1985);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,891,675,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=a^2*c^3>;
 
// generators/relations
 

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