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G = C23.51D8order 128 = 27

22nd non-split extension by C23 of D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.51D8, C22.3Q32, C163C48C2, (C2×C8).76D4, (C2×C4).44D8, C2.8(C2×Q32), C2.Q327C2, C8.73(C4○D4), C22⋊C16.6C2, (C2×C8).536C23, (C2×C16).11C22, C8.18D4.6C2, C22.122(C2×D8), (C22×C4).356D4, C2.19(C16⋊C22), C2.D8.21C22, C4.18(C8.C22), (C2×Q16).11C22, (C22×C8).132C22, C4.43(C22.D4), C2.16(C22.D8), (C2×C4).804(C2×D4), (C2×C2.D8).26C2, SmallGroup(128,968)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.51D8
C1C2C4C8C2×C8C2.D8C2×C2.D8 — C23.51D8
C1C2C4C2×C8 — C23.51D8
C1C22C22×C4C22×C8 — C23.51D8
C1C2C2C2C2C4C4C2×C8 — C23.51D8

Generators and relations for C23.51D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd7 >

Subgroups: 164 in 70 conjugacy classes, 32 normal (20 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C8 [×2], C8, C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, C16 [×2], C22⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], Q16 [×2], C22×C4, C22×C4, C2×Q8, Q8⋊C4, C2.D8, C2.D8 [×2], C2.D8, C2×C16 [×2], C2×C4⋊C4, C22⋊Q8, C22×C8, C2×Q16, C22⋊C16, C2.Q32 [×2], C163C4 [×2], C2×C2.D8, C8.18D4, C23.51D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, C4○D4 [×2], Q32 [×2], C22.D4, C2×D8, C8.C22, C22.D8, C2×Q32, C16⋊C22, C23.51D8

Character table of C23.51D8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111222248888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-111-111-1-1-111111-1-11-1-1-1111-1    linear of order 2
ρ41111-1-111-111-1-11-11111-1-1-1111-1-1-11    linear of order 2
ρ5111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ6111111111-1-1-1-1-1-111111111111111    linear of order 2
ρ71111-1-111-1-1-111-111111-1-1-1111-1-1-11    linear of order 2
ρ81111-1-111-1-1-1111-11111-1-11-1-1-1111-1    linear of order 2
ρ92222-2-222-2000000-2-2-2-22200000000    orthogonal lifted from D4
ρ10222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ11222222-2-2-200000000000022-22-22-2-2    orthogonal lifted from D8
ρ12222222-2-2-2000000000000-2-22-22-222    orthogonal lifted from D8
ρ132222-2-2-2-22000000000000-22-222-22-2    orthogonal lifted from D8
ρ142222-2-2-2-220000000000002-22-2-22-22    orthogonal lifted from D8
ρ152-22-22-200000000022-2-2-22ζ1671616716165163ζ1671616516316716ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ162-22-22-200000000022-2-2-2216716ζ16716ζ16516316716ζ165163ζ16716165163165163    symplectic lifted from Q32, Schur index 2
ρ172-22-22-2000000000-2-2222-2ζ165163165163ζ16716ζ165163ζ167161651631671616716    symplectic lifted from Q32, Schur index 2
ρ182-22-2-22000000000-2-222-22165163165163ζ16716ζ16516316716ζ165163ζ1671616716    symplectic lifted from Q32, Schur index 2
ρ192-22-22-2000000000-2-2222-2165163ζ1651631671616516316716ζ165163ζ16716ζ16716    symplectic lifted from Q32, Schur index 2
ρ202-22-2-22000000000-2-222-22ζ165163ζ16516316716165163ζ1671616516316716ζ16716    symplectic lifted from Q32, Schur index 2
ρ212-22-2-2200000000022-2-22-21671616716165163ζ16716ζ165163ζ16716165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ222-22-2-2200000000022-2-22-2ζ16716ζ16716ζ1651631671616516316716ζ165163165163    symplectic lifted from Q32, Schur index 2
ρ232-2-22002-2000-2i2i00-22-220000000000    complex lifted from C4○D4
ρ242-2-22002-202i-2i00002-22-20000000000    complex lifted from C4○D4
ρ252-2-22002-20-2i2i00002-22-20000000000    complex lifted from C4○D4
ρ262-2-22002-20002i-2i00-22-220000000000    complex lifted from C4○D4
ρ2744-4-40000000000022-22-22220000000000    orthogonal lifted from C16⋊C22
ρ2844-4-400000000000-222222-220000000000    orthogonal lifted from C16⋊C22
ρ294-4-4400-44000000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C23.51D8
On 64 points
Generators in S64
(2 56)(4 58)(6 60)(8 62)(10 64)(12 50)(14 52)(16 54)(18 48)(20 34)(22 36)(24 38)(26 40)(28 42)(30 44)(32 46)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 47)(18 48)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(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 22 9 30)(2 35 10 43)(3 20 11 28)(4 33 12 41)(5 18 13 26)(6 47 14 39)(7 32 15 24)(8 45 16 37)(17 52 25 60)(19 50 27 58)(21 64 29 56)(23 62 31 54)(34 49 42 57)(36 63 44 55)(38 61 46 53)(40 59 48 51)

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

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

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

Matrix representation of C23.51D8 in GL4(𝔽17) generated by

1000
0100
0010
00016
,
1000
0100
00160
00016
,
16000
01600
0010
0001
,
61300
4600
00013
00130
,
10100
1700
0001
0010
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[6,4,0,0,13,6,0,0,0,0,0,13,0,0,13,0],[10,1,0,0,1,7,0,0,0,0,0,1,0,0,1,0] >;

C23.51D8 in GAP, Magma, Sage, TeX

C_2^3._{51}D_8
% in TeX

G:=Group("C2^3.51D8");
// GroupNames label

G:=SmallGroup(128,968);
// by ID

G=gap.SmallGroup(128,968);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,394,1684,438,242,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^7>;
// generators/relations

Export

Character table of C23.51D8 in TeX

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