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G = C42.87D4order 128 = 27

69th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.87D4, C42.178C23, C4.D826C2, C4⋊C8.212C22, C4.87(C8⋊C22), C42.119(C2×C4), C4.6Q1626C2, C4.4D4.12C4, (C22×C4).247D4, C4⋊Q8.250C22, C4⋊M4(2)⋊22C2, C4.17(C4.D4), C4.89(C8.C22), C41D4.132C22, C23.68(C22⋊C4), (C2×C42).222C22, C2.18(C23.36D4), C22.26C24.18C2, (C2×C4○D4).8C4, (C2×D4).36(C2×C4), (C2×Q8).36(C2×C4), (C2×C4).1249(C2×D4), C2.23(C2×C4.D4), (C2×C4).172(C22×C4), (C22×C4).244(C2×C4), (C2×C4).109(C22⋊C4), C22.236(C2×C22⋊C4), SmallGroup(128,292)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.87D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.87D4
C1C22C2×C4 — C42.87D4
C1C22C2×C42 — C42.87D4
C1C22C22C42 — C42.87D4

Generators and relations for C42.87D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 292 in 127 conjugacy classes, 48 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×6], C4 [×5], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C4.D8 [×2], C4.6Q16 [×2], C4⋊M4(2) [×2], C22.26C24, C42.87D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C2×C22⋊C4, C8⋊C22 [×2], C8.C22 [×2], C2×C4.D4, C23.36D4 [×2], C42.87D4

Character table of C42.87D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 11114882222224448888888888
ρ111111111111111111111111111    trivial
ρ21111-1111-11-111-11-1-1-1-1-1-1111-11    linear of order 2
ρ31111-1-1-11-11-111-11-1111-1-11-1-111    linear of order 2
ρ411111-1-1111111111-1-1-1111-1-1-11    linear of order 2
ρ51111-1111-11-111-11-1-1-1111-1-1-11-1    linear of order 2
ρ6111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ711111-1-1111111111-1-11-1-1-1111-1    linear of order 2
ρ81111-1-1-11-11-111-11-111-111-111-1-1    linear of order 2
ρ911111-11-11-11-1-1-11-11-1ii-i-i-ii-ii    linear of order 4
ρ101111-1-11-1-1-1-1-1-1111-11-i-ii-i-iiii    linear of order 4
ρ11111111-1-11-11-1-1-11-1-11-ii-i-ii-iii    linear of order 4
ρ121111-11-1-1-1-1-1-1-11111-1i-ii-ii-i-ii    linear of order 4
ρ131111-1-11-1-1-1-1-1-1111-11ii-iii-i-i-i    linear of order 4
ρ1411111-11-11-11-1-1-11-11-1-i-iiii-ii-i    linear of order 4
ρ151111-11-1-1-1-1-1-1-11111-1-ii-ii-iii-i    linear of order 4
ρ16111111-1-11-11-1-1-11-1-11i-iii-ii-i-i    linear of order 4
ρ1722222002-2-2-2-22-2-220000000000    orthogonal lifted from D4
ρ182222-20022-22-222-2-20000000000    orthogonal lifted from D4
ρ192222-200-22222-2-2-220000000000    orthogonal lifted from D4
ρ202222200-2-22-22-22-2-20000000000    orthogonal lifted from D4
ρ214-4-440000-404000000000000000    orthogonal lifted from C4.D4
ρ224-4-44000040-4000000000000000    orthogonal lifted from C4.D4
ρ234-44-400040000-40000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-40000040-400000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-400000-40400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-4000-4000040000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.87D4
On 64 points
Generators in S64
(1 13 61 25)(2 26 62 14)(3 15 63 27)(4 28 64 16)(5 9 57 29)(6 30 58 10)(7 11 59 31)(8 32 60 12)(17 56 37 41)(18 42 38 49)(19 50 39 43)(20 44 40 51)(21 52 33 45)(22 46 34 53)(23 54 35 47)(24 48 36 55)
(1 18 57 34)(2 35 58 19)(3 20 59 36)(4 37 60 21)(5 22 61 38)(6 39 62 23)(7 24 63 40)(8 33 64 17)(9 46 25 49)(10 50 26 47)(11 48 27 51)(12 52 28 41)(13 42 29 53)(14 54 30 43)(15 44 31 55)(16 56 32 45)
(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 64 18 17 57 8 34 33)(2 40 35 7 58 24 19 63)(3 62 20 23 59 6 36 39)(4 38 37 5 60 22 21 61)(9 32 46 45 25 16 49 56)(10 55 50 15 26 44 47 31)(11 30 48 43 27 14 51 54)(12 53 52 13 28 42 41 29)

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

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

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

Matrix representation of C42.87D4 in GL8(𝔽17)

10000000
01000000
00100000
00010000
000010150
000001015
000010160
000001016
,
01000000
160000000
1161150000
101160000
00001000
00000100
00000010
00000001
,
0130130000
004130000
4134130000
4130130000
0000611710
00006677
0000116116
0000111111
,
004130000
0130130000
130040000
134040000
0000611710
000011111010
0000116116
0000161666

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

C42.87D4 in GAP, Magma, Sage, TeX

C_4^2._{87}D_4
% in TeX

G:=Group("C4^2.87D4");
// GroupNames label

G:=SmallGroup(128,292);
// by ID

G=gap.SmallGroup(128,292);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,1018,248,1971,242]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

Export

Character table of C42.87D4 in TeX

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