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G = C42.88D4order 128 = 27

70th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.88D4, C42.179C23, C4.10D842C2, C4⋊C8.213C22, C4.88(C8⋊C22), C42.120(C2×C4), C42⋊C2.8C4, (C22×C4).248D4, C4⋊Q8.251C22, C42.C2.14C4, C4.90(C8.C22), C4.17(C4.10D4), C4⋊M4(2).19C2, C23.69(C22⋊C4), (C2×C42).223C22, C2.19(C23.36D4), C23.37C23.18C2, C4⋊C4.46(C2×C4), (C2×C4).1250(C2×D4), (C22×C4).245(C2×C4), (C2×C4).173(C22×C4), C2.21(C2×C4.10D4), (C2×C4).110(C22⋊C4), C22.237(C2×C22⋊C4), SmallGroup(128,293)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.88D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.88D4
C1C22C2×C4 — C42.88D4
C1C22C2×C42 — C42.88D4
C1C22C22C42 — C42.88D4

Generators and relations for C42.88D4
 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=bc3 >

Subgroups: 196 in 105 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×6], C4 [×7], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], C23, C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C4.10D8 [×4], C4⋊M4(2) [×2], C23.37C23, C42.88D4
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, C23.36D4 [×2], C42.88D4

Character table of C42.88D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F8G8H
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-11-1-1111    linear of order 2
ρ311111111111111-1-1-1-111-111-1-1-1    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-111-1-111-1-11-111-1-111-11-1-11    linear of order 2
ρ61111-111-1-111-1-111-1-111-111-1-1-11    linear of order 2
ρ71111-111-1-111-1-111-1-11-11-1-1111-1    linear of order 2
ρ81111-111-1-111-1-11-111-11-1-11-111-1    linear of order 2
ρ911111-1-111-1-1-1-1111-1-1ii-i-i-i-iii    linear of order 4
ρ1011111-1-111-1-1-1-11-1-111-i-i-iii-iii    linear of order 4
ρ1111111-1-111-1-1-1-1111-1-1-i-iiiii-i-i    linear of order 4
ρ1211111-1-111-1-1-1-11-1-111iii-i-ii-i-i    linear of order 4
ρ131111-1-1-1-1-1-1-1111-11-11-ii-ii-ii-ii    linear of order 4
ρ141111-1-1-1-1-1-1-11111-11-1i-i-i-iii-ii    linear of order 4
ρ151111-1-1-1-1-1-1-1111-11-11i-ii-ii-ii-i    linear of order 4
ρ161111-1-1-1-1-1-1-11111-11-1-iiii-i-ii-i    linear of order 4
ρ172222-2-2-222222-2-2000000000000    orthogonal lifted from D4
ρ182222-22222-2-2-22-2000000000000    orthogonal lifted from D4
ρ192222222-2-2-2-22-2-2000000000000    orthogonal lifted from D4
ρ2022222-2-2-2-222-22-2000000000000    orthogonal lifted from D4
ρ214-44-404-40000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-4400000-44000000000000000    orthogonal lifted from C8⋊C22
ρ234-4-44000004-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2444-4-4000-4400000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ254-44-40-440000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-40004-400000000000000000    symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C42.88D4
On 64 points
Generators in S64
(1 59 23 9)(2 10 24 60)(3 61 17 11)(4 12 18 62)(5 63 19 13)(6 14 20 64)(7 57 21 15)(8 16 22 58)(25 49 46 34)(26 35 47 50)(27 51 48 36)(28 37 41 52)(29 53 42 38)(30 39 43 54)(31 55 44 40)(32 33 45 56)
(1 15 19 61)(2 62 20 16)(3 9 21 63)(4 64 22 10)(5 11 23 57)(6 58 24 12)(7 13 17 59)(8 60 18 14)(25 51 42 40)(26 33 43 52)(27 53 44 34)(28 35 45 54)(29 55 46 36)(30 37 47 56)(31 49 48 38)(32 39 41 50)
(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 15 38 19 31 61 49)(2 52 62 26 20 33 16 43)(3 46 9 36 21 29 63 55)(4 50 64 32 22 39 10 41)(5 44 11 34 23 27 57 53)(6 56 58 30 24 37 12 47)(7 42 13 40 17 25 59 51)(8 54 60 28 18 35 14 45)

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

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

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

Matrix representation of C42.88D4 in GL8(𝔽17)

40000000
04000000
001300000
000130000
00001200
0000161600
00000012
0000001616
,
10000000
01000000
00100000
00010000
00001200
0000161600
0000001615
00000011
,
00010000
001600000
013000000
130000000
000013900
00000400
00000040
0000001313
,
00100000
00010000
10000000
01000000
00000040
0000001313
000013900
00000400

G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0] >;

C42.88D4 in GAP, Magma, Sage, TeX

C_4^2._{88}D_4
% in TeX

G:=Group("C4^2.88D4");
// GroupNames label

G:=SmallGroup(128,293);
// by ID

G=gap.SmallGroup(128,293);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,352,1123,1018,248,1971,242]);
// 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=b*c^3>;
// generators/relations

Export

Character table of C42.88D4 in TeX

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