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G = C42.83D4order 128 = 27

65th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.83D4, C42.174C23, C4⋊Q8.25C4, C4.10D839C2, C4⋊C8.208C22, C42.115(C2×C4), (C22×C4).243D4, C4⋊Q8.246C22, C4.112(C8⋊C22), C4.107(C8.C22), C42.6C4.25C2, (C2×C42).218C22, C23.186(C22⋊C4), C22.11(C4.10D4), C2.14(C23.38D4), C2.15(C23.37D4), (C2×C4⋊Q8).6C2, (C2×C4⋊C4).22C4, C4⋊C4.43(C2×C4), (C2×C4).1245(C2×D4), (C2×C4).168(C22×C4), (C22×C4).240(C2×C4), C2.20(C2×C4.10D4), (C2×C4).107(C22⋊C4), C22.232(C2×C22⋊C4), SmallGroup(128,288)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.83D4
C1C2C22C2×C4C42C2×C42C2×C4⋊Q8 — C42.83D4
C1C22C2×C4 — C42.83D4
C1C22C2×C42 — C42.83D4
C1C22C22C42 — C42.83D4

Generators and relations for C42.83D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=bc3 >

Subgroups: 228 in 114 conjugacy classes, 48 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×14], Q8 [×8], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×8], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C4.10D8 [×4], C42.6C4 [×2], C2×C4⋊Q8, C42.83D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C8⋊C22 [×2], C8.C22 [×2], C2×C4.10D4, C23.37D4, C23.38D4, C42.83D4

Character table of C42.83D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111-1-111111-1-1-111-1-11-1-111-1-11    linear of order 2
ρ31111-1-111111-1-1-1-1-111-11-11-11-11    linear of order 2
ρ411111111111111-1-1-1-1-1-111-1-111    linear of order 2
ρ511111111111111-1-1-1-111-1-111-1-1    linear of order 2
ρ61111-1-111111-1-1-1-1-1111-11-11-11-1    linear of order 2
ρ71111-1-111111-1-1-111-1-1-111-1-111-1    linear of order 2
ρ8111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91111-1-1-1-1-1-111-111-1-11i-ii-i-ii-ii    linear of order 4
ρ10111111-1-1-1-11-11-11-11-1ii-i-i-i-iii    linear of order 4
ρ11111111-1-1-1-11-11-1-11-11-i-i-i-iiiii    linear of order 4
ρ121111-1-1-1-1-1-111-11-111-1-iii-ii-i-ii    linear of order 4
ρ13111111-1-1-1-11-11-11-11-1-i-iiiii-i-i    linear of order 4
ρ141111-1-1-1-1-1-111-111-1-11-ii-iii-ii-i    linear of order 4
ρ151111-1-1-1-1-1-111-11-111-1i-i-ii-iii-i    linear of order 4
ρ16111111-1-1-1-11-11-1-11-11iiii-i-i-i-i    linear of order 4
ρ172222-2-2-2-222-2-222000000000000    orthogonal lifted from D4
ρ182222-2-222-2-2-222-2000000000000    orthogonal lifted from D4
ρ19222222-2-222-22-2-2000000000000    orthogonal lifted from D4
ρ2022222222-2-2-2-2-22000000000000    orthogonal lifted from D4
ρ2144-4-400-44000000000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4004-4000000000000000000    orthogonal lifted from C8⋊C22
ρ234-44-44-400000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ244-44-4-4400000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ254-4-4400004-40000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-440000-440000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.83D4
On 64 points
Generators in S64
(1 63 54 35)(2 60 55 40)(3 57 56 37)(4 62 49 34)(5 59 50 39)(6 64 51 36)(7 61 52 33)(8 58 53 38)(9 23 27 43)(10 20 28 48)(11 17 29 45)(12 22 30 42)(13 19 31 47)(14 24 32 44)(15 21 25 41)(16 18 26 46)
(1 61 50 37)(2 38 51 62)(3 63 52 39)(4 40 53 64)(5 57 54 33)(6 34 55 58)(7 59 56 35)(8 36 49 60)(9 21 31 45)(10 46 32 22)(11 23 25 47)(12 48 26 24)(13 17 27 41)(14 42 28 18)(15 19 29 43)(16 44 30 20)
(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 61 26 50 24 37 12)(2 15 38 19 51 29 62 43)(3 46 63 32 52 22 39 10)(4 13 40 17 53 27 64 41)(5 44 57 30 54 20 33 16)(6 11 34 23 55 25 58 47)(7 42 59 28 56 18 35 14)(8 9 36 21 49 31 60 45)

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

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

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

Matrix representation of C42.83D4 in GL8(𝔽17)

1615000000
11000000
13130160000
04100000
00004000
000001300
00000840
000020013
,
1615000000
11000000
013010000
441600000
00001000
00000100
00000010
00000001
,
1006110000
120060000
631250000
1435120000
0000915015
00002220
0000615152
0000116158
,
401500000
00110000
011300000
10400000
00001515150
0000915015
00001422
000016182

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

C42.83D4 in GAP, Magma, Sage, TeX

C_4^2._{83}D_4
% in TeX

G:=Group("C4^2.83D4");
// GroupNames label

G:=SmallGroup(128,288);
// by ID

G=gap.SmallGroup(128,288);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,1018,248,1971,242]);
// 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=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations

Export

Character table of C42.83D4 in TeX

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