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G = C42.84D4order 128 = 27

66th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.84D4, C42.175C23, (C4×D4).9C4, C4⋊D4.16C4, C4.D825C2, C4.10D840C2, C4⋊C8.209C22, C42.116(C2×C4), (C22×C4).244D4, C4⋊Q8.247C22, C4.113(C8⋊C22), C4⋊M4(2)⋊21C2, C42.6C443C2, C4.88(C8.C22), C41D4.131C22, C23.66(C22⋊C4), (C2×C42).219C22, C2.16(C23.37D4), C2.16(C23.36D4), C22.26C24.17C2, C2.20(M4(2).8C22), C4⋊C4.44(C2×C4), (C2×D4).35(C2×C4), (C2×C4).1246(C2×D4), (C2×C4).169(C22×C4), (C22×C4).241(C2×C4), (C2×C4).323(C22⋊C4), C22.233(C2×C22⋊C4), SmallGroup(128,289)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.84D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.84D4
C1C22C2×C4 — C42.84D4
C1C22C2×C42 — C42.84D4
C1C22C22C42 — C42.84D4

Generators and relations for C42.84D4
 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=a2bc3 >

Subgroups: 284 in 123 conjugacy classes, 46 normal (28 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×11], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C8⋊C4, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×M4(2), C2×C4○D4, C4.D8 [×2], C4.10D8 [×2], C4⋊M4(2), C42.6C4, C22.26C24, C42.84D4
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 [×3], C8.C22, M4(2).8C22, C23.36D4, C23.37D4, C42.84D4

Character table of C42.84D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 11114882222224448888888888
ρ111111111111111111111111111    trivial
ρ21111-1111-111-11-11-1-1-1-111-1-11-11    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1111-111-11-11-1-1-11-1-111-11-1    linear of order 2
ρ51111-1-1-11-111-11-11-1111-1-11-11-11    linear of order 2
ρ611111-1-1111111111-1-1-1-1-1-11111    linear of order 2
ρ71111-1-1-11-111-11-11-111-111-11-11-1    linear of order 2
ρ811111-1-1111111111-1-11111-1-1-1-1    linear of order 2
ρ91111-11-1-11-1-11-111-1-11-i-iiiii-i-i    linear of order 4
ρ10111111-1-1-1-1-1-1-1-1111-1i-ii-i-iii-i    linear of order 4
ρ111111-11-1-11-1-11-111-1-11ii-i-i-i-iii    linear of order 4
ρ12111111-1-1-1-1-1-1-1-1111-1-ii-iii-i-ii    linear of order 4
ρ1311111-11-1-1-1-1-1-1-111-11-ii-ii-iii-i    linear of order 4
ρ141111-1-11-11-1-11-111-11-1ii-i-iii-i-i    linear of order 4
ρ1511111-11-1-1-1-1-1-1-111-11i-ii-ii-i-ii    linear of order 4
ρ161111-1-11-11-1-11-111-11-1-i-iii-i-iii    linear of order 4
ρ17222220022-222-2-2-2-20000000000    orthogonal lifted from D4
ρ182222-2002-2-22-2-22-220000000000    orthogonal lifted from D4
ρ192222-200-222-222-2-220000000000    orthogonal lifted from D4
ρ202222200-2-22-2-222-2-20000000000    orthogonal lifted from D4
ρ214-4-44000400-4000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44000-4004000000000000000    orthogonal lifted from C8⋊C22
ρ2344-4-400000400-40000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400000-40040000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-44-40000-4i004i00000000000000    complex lifted from M4(2).8C22
ρ264-44-400004i00-4i00000000000000    complex lifted from M4(2).8C22

Smallest permutation representation of C42.84D4
On 64 points
Generators in S64
(1 45 5 41)(2 50 6 54)(3 47 7 43)(4 52 8 56)(9 57 13 61)(10 30 14 26)(11 59 15 63)(12 32 16 28)(17 42 21 46)(18 55 22 51)(19 44 23 48)(20 49 24 53)(25 40 29 36)(27 34 31 38)(33 58 37 62)(35 60 39 64)
(1 25 24 57)(2 58 17 26)(3 27 18 59)(4 60 19 28)(5 29 20 61)(6 62 21 30)(7 31 22 63)(8 64 23 32)(9 41 36 49)(10 50 37 42)(11 43 38 51)(12 52 39 44)(13 45 40 53)(14 54 33 46)(15 47 34 55)(16 56 35 48)
(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 25 23 24 32 57 8)(2 7 58 31 17 22 26 63)(3 62 27 21 18 30 59 6)(4 5 60 29 19 20 28 61)(9 48 41 16 36 56 49 35)(10 34 50 55 37 15 42 47)(11 46 43 14 38 54 51 33)(12 40 52 53 39 13 44 45)

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

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

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

Matrix representation of C42.84D4 in GL8(𝔽17)

00100000
00010000
10000000
01000000
000013000
000001300
000000130
000000013
,
01000000
160000000
00010000
001600000
0000161600
00002100
0000011615
00001011
,
003140000
0014140000
143000000
33000000
0000101112
00001010712
000060165
0000101077
,
003140000
00330000
143000000
1414000000
0000101112
000057515
000060165
000007010

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

C42.84D4 in GAP, Magma, Sage, TeX

C_4^2._{84}D_4
% in TeX

G:=Group("C4^2.84D4");
// GroupNames label

G:=SmallGroup(128,289);
// by ID

G=gap.SmallGroup(128,289);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,1018,248,1971,242]);
// 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=a^2*b*c^3>;
// generators/relations

Export

Character table of C42.84D4 in TeX

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