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G = C42.71D4order 128 = 27

53rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.71D4, C42.152C23, C42.93(C2×C4), (C22×C4).229D4, C42.C2.11C4, C8⋊C4.87C22, C42.6C4.21C2, (C2×C42).196C22, C42.C2.98C22, C23.180(C22⋊C4), C42.2C2210C2, C2.34(C42⋊C22), C22.10(C4.10D4), (C2×C4⋊C4).18C4, C4⋊C4.29(C2×C4), (C2×C4).1180(C2×D4), (C2×C42.C2).4C2, (C2×C4).97(C22⋊C4), (C2×C4).146(C22×C4), (C22×C4).218(C2×C4), C2.13(C2×C4.10D4), C22.210(C2×C22⋊C4), SmallGroup(128,266)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.71D4
C1C2C22C2×C4C42C2×C42C2×C42.C2 — C42.71D4
C1C22C2×C4 — C42.71D4
C1C22C2×C42 — C42.71D4
C1C22C22C42 — C42.71D4

Generators and relations for C42.71D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >

Subgroups: 180 in 98 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], C23, C42 [×2], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C42.2C22 [×4], C42.6C4 [×2], C2×C42.C2, C42.71D4
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, C2×C4.10D4, C42⋊C22 [×2], C42.71D4

Character table of C42.71D4

 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
ρ214-44-44-400000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ224-44-4-4400000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ2344-4-4004i-4i000000000000000000    complex lifted from C42⋊C22
ρ244-4-440000-4i4i0000000000000000    complex lifted from C42⋊C22
ρ2544-4-400-4i4i000000000000000000    complex lifted from C42⋊C22
ρ264-4-4400004i-4i0000000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C42.71D4
On 64 points
Generators in S64
(1 62 49 38)(2 35 50 59)(3 64 51 40)(4 37 52 61)(5 58 53 34)(6 39 54 63)(7 60 55 36)(8 33 56 57)(9 48 29 20)(10 17 30 45)(11 42 31 22)(12 19 32 47)(13 44 25 24)(14 21 26 41)(15 46 27 18)(16 23 28 43)
(1 36 53 64)(2 33 54 61)(3 38 55 58)(4 35 56 63)(5 40 49 60)(6 37 50 57)(7 34 51 62)(8 39 52 59)(9 46 25 22)(10 43 26 19)(11 48 27 24)(12 45 28 21)(13 42 29 18)(14 47 30 23)(15 44 31 20)(16 41 32 17)
(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 20 36 15 53 44 64 31)(2 26 33 19 54 10 61 43)(3 46 38 25 55 22 58 9)(4 12 35 45 56 28 63 21)(5 24 40 11 49 48 60 27)(6 30 37 23 50 14 57 47)(7 42 34 29 51 18 62 13)(8 16 39 41 52 32 59 17)

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

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

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

Matrix representation of C42.71D4 in GL8(𝔽17)

04000000
40000000
000130000
001300000
0000161600
00002100
0000001616
00000021
,
01000000
10000000
00010000
00100000
00004000
00000400
00000040
00000004
,
00470000
0010130000
16000000
1116000000
000000712
0000001010
000001400
000011000
,
00100000
00010000
01000000
10000000
00000010
00000001
00004000
00000400

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

C42.71D4 in GAP, Magma, Sage, TeX

C_4^2._{71}D_4
% in TeX

G:=Group("C4^2.71D4");
// GroupNames label

G:=SmallGroup(128,266);
// by ID

G=gap.SmallGroup(128,266);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,1018,248,1971,102]);
// 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*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations

Export

Character table of C42.71D4 in TeX

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