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G = C42.86D4order 128 = 27

68th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.86D4, C42.177C23, (C4×Q8).9C4, C22⋊Q8.16C4, C4.10D841C2, C4⋊C8.211C22, C4.86(C8⋊C22), C42.118(C2×C4), C4.6Q1625C2, (C22×C4).246D4, C4⋊Q8.249C22, C4.109(C8.C22), C4⋊M4(2).18C2, C23.67(C22⋊C4), C42.6C4.27C2, (C2×C42).221C22, C2.16(C23.38D4), C2.17(C23.36D4), C23.37C23.17C2, C2.21(M4(2).8C22), C4⋊C4.45(C2×C4), (C2×Q8).35(C2×C4), (C2×C4).1248(C2×D4), (C2×C4).171(C22×C4), (C22×C4).243(C2×C4), (C2×C4).324(C22⋊C4), C22.235(C2×C22⋊C4), SmallGroup(128,291)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.86D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.86D4
C1C22C2×C4 — C42.86D4
C1C22C2×C42 — C42.86D4
C1C22C22C42 — C42.86D4

Generators and relations for C42.86D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 188 in 101 conjugacy classes, 46 normal (28 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×8], C22, C22 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C2×Q8 [×2], C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C2×M4(2), C4.10D8 [×2], C4.6Q16 [×2], C4⋊M4(2), C42.6C4, C23.37C23, C42.86D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8⋊C22, C8.C22 [×3], M4(2).8C22, C23.36D4, C23.38D4, C42.86D4

Character table of C42.86D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F8G8H
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ21111-1111-1-111-1-1-111-11-1-111-11-1    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ41111-1111-1-111-1-11-1-11-111-11-11-1    linear of order 2
ρ51111-1111-1-111-1-1-111-1-111-1-11-11    linear of order 2
ρ6111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1111-1-111-1-11-1-111-1-11-11-11    linear of order 2
ρ811111111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ911111-1-1-1-1-1-111-1-11-11-ii-ii-iii-i    linear of order 4
ρ101111-1-1-1-111-11-1111-1-1-i-iii-i-iii    linear of order 4
ρ1111111-1-1-1-1-1-111-11-11-1i-ii-i-iii-i    linear of order 4
ρ121111-1-1-1-111-11-11-1-111ii-i-i-i-iii    linear of order 4
ρ131111-1-1-1-111-11-1111-1-1ii-i-iii-i-i    linear of order 4
ρ1411111-1-1-1-1-1-111-1-11-11i-ii-ii-i-ii    linear of order 4
ρ151111-1-1-1-111-11-11-1-111-i-iiiii-i-i    linear of order 4
ρ1611111-1-1-1-1-1-111-11-11-1-ii-iii-i-ii    linear of order 4
ρ1722222-22-2-2-22-2-22000000000000    orthogonal lifted from D4
ρ182222-2-22-2222-22-2000000000000    orthogonal lifted from D4
ρ19222222-2222-2-2-2-2000000000000    orthogonal lifted from D4
ρ202222-22-22-2-2-2-222000000000000    orthogonal lifted from D4
ρ2144-4-4004000-4000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44040-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-4-440-404000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2444-4-400-40004000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-44-40000-4i4i0000000000000000    complex lifted from M4(2).8C22
ρ264-44-400004i-4i0000000000000000    complex lifted from M4(2).8C22

Smallest permutation representation of C42.86D4
On 64 points
Generators in S64
(1 7 5 3)(2 61 6 57)(4 63 8 59)(9 26 13 30)(10 12 14 16)(11 28 15 32)(17 19 21 23)(18 44 22 48)(20 46 24 42)(25 27 29 31)(33 39 37 35)(34 49 38 53)(36 51 40 55)(41 43 45 47)(50 56 54 52)(58 64 62 60)
(1 52 58 39)(2 40 59 53)(3 54 60 33)(4 34 61 55)(5 56 62 35)(6 36 63 49)(7 50 64 37)(8 38 57 51)(9 24 28 44)(10 45 29 17)(11 18 30 46)(12 47 31 19)(13 20 32 48)(14 41 25 21)(15 22 26 42)(16 43 27 23)
(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 48 52 13 58 20 39 32)(2 16 40 43 59 27 53 23)(3 46 54 11 60 18 33 30)(4 14 34 41 61 25 55 21)(5 44 56 9 62 24 35 28)(6 12 36 47 63 31 49 19)(7 42 50 15 64 22 37 26)(8 10 38 45 57 29 51 17)

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

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

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

Matrix representation of C42.86D4 in GL8(𝔽17)

40000000
04000000
00400000
00040000
000011500
000011600
0000142115
0000153116
,
115000000
116000000
1610160000
01100000
000016000
000001600
000000160
000000016
,
14000000
316000000
1515950000
615580000
00001514126
0000156155
000014111114
000073152
,
301670000
801330000
1515950000
114950000
00009200
000010800
0000591114
000011516

G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,14,15,0,0,0,0,15,16,2,3,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[1,1,16,0,0,0,0,0,15,16,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,3,15,6,0,0,0,0,4,16,15,15,0,0,0,0,0,0,9,5,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,15,15,14,7,0,0,0,0,14,6,11,3,0,0,0,0,12,15,11,15,0,0,0,0,6,5,14,2],[3,8,15,1,0,0,0,0,0,0,15,14,0,0,0,0,16,13,9,9,0,0,0,0,7,3,5,5,0,0,0,0,0,0,0,0,9,10,5,1,0,0,0,0,2,8,9,15,0,0,0,0,0,0,11,1,0,0,0,0,0,0,14,6] >;

C42.86D4 in GAP, Magma, Sage, TeX

C_4^2._{86}D_4
% in TeX

G:=Group("C4^2.86D4");
// GroupNames label

G:=SmallGroup(128,291);
// by ID

G=gap.SmallGroup(128,291);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,1430,184,1123,1018,248,1971,242]);
// Polycyclic

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

Export

Character table of C42.86D4 in TeX

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