Copied to
clipboard

G = C23.20D8order 128 = 27

13rd non-split extension by C23 of D8 acting via D8/C4=C22

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

Aliases: C23.20D8, C163C49C2, C164C412C2, (C2×C8).179D4, (C2×C4).115D8, C8.74(C4○D4), C22⋊C16.7C2, C2.Q3213C2, C2.16(C4○D16), (C2×C16).12C22, (C2×C8).537C23, C8.18D4.9C2, (C22×C4).357D4, C22.123(C2×D8), C2.D8.22C22, C2.19(Q32⋊C2), C4.19(C8.C22), (C2×Q16).12C22, (C22×C8).175C22, C23.25D4.5C2, C4.44(C22.D4), C2.17(C22.D8), (C2×C4).805(C2×D4), SmallGroup(128,969)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.20D8
C1C2C4C8C2×C8C2.D8C23.25D4 — C23.20D8
C1C2C4C2×C8 — C23.20D8
C1C22C22×C4C22×C8 — C23.20D8
C1C2C2C2C2C4C4C2×C8 — C23.20D8

Generators and relations for C23.20D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd7 >

Subgroups: 148 in 66 conjugacy classes, 30 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×2], C8, C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C16 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], Q16 [×2], C22×C4, C2×Q8, Q8⋊C4, C4.Q8, C2.D8 [×3], C2×C16 [×2], C42⋊C2, C22⋊Q8, C22×C8, C2×Q16, C22⋊C16, C2.Q32 [×2], C163C4, C164C4, C23.25D4, C8.18D4, C23.20D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, C4○D4 [×2], C22.D4, C2×D8, C8.C22, C22.D8, C4○D16, Q32⋊C2, C23.20D8

Character table of C23.20D8

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111422228888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-11-1-1111-1-1-111111-1-1-1-1-111-111    linear of order 2
ρ41111-11-1-1111-1-11-11111-1-1111-1-11-1-1    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
ρ71111-11-1-11-1-111-111111-1-1111-1-11-1-1    linear of order 2
ρ81111-11-1-11-1-1111-11111-1-1-1-1-111-111    linear of order 2
ρ9222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ102222-22-2-22000000-2-2-2-22200000000    orthogonal lifted from D4
ρ1122222-2-2-2-20000000000002-222-2-2-22    orthogonal lifted from D8
ρ1222222-2-2-2-2000000000000-22-2-2222-2    orthogonal lifted from D8
ρ132222-2-222-20000000000002-22-22-22-2    orthogonal lifted from D8
ρ142222-2-222-2000000000000-22-22-22-22    orthogonal lifted from D8
ρ152-2-220200-200-2i2i002-2-220000000000    complex lifted from C4○D4
ρ162-2-220200-2002i-2i002-2-220000000000    complex lifted from C4○D4
ρ172-2-220200-2-2i2i0000-222-20000000000    complex lifted from C4○D4
ρ182-2-220200-22i-2i0000-222-20000000000    complex lifted from C4○D4
ρ1922-2-2002i-2i0000000-22-22--2-2ζ16716ζ16516316716ζ1615169ζ16131611165163ζ165163ζ16716    complex lifted from C4○D16
ρ2022-2-200-2i2i0000000-22-22-2--216716165163ζ16716ζ1615169ζ16131611ζ165163ζ165163ζ16716    complex lifted from C4○D16
ρ2122-2-200-2i2i0000000-22-22-2--2ζ16716ζ16516316716ζ16716ζ165163165163ζ16131611ζ1615169    complex lifted from C4○D16
ρ2222-2-2002i-2i0000000-22-22--2-216716165163ζ16716ζ16716ζ165163ζ165163ζ16131611ζ1615169    complex lifted from C4○D16
ρ2322-2-2002i-2i00000002-22-2-2--2ζ16516316716165163ζ165163ζ1615169ζ16716ζ16716ζ16131611    complex lifted from C4○D16
ρ2422-2-2002i-2i00000002-22-2-2--2165163ζ16716ζ165163ζ16131611ζ1671616716ζ1615169ζ165163    complex lifted from C4○D16
ρ2522-2-200-2i2i00000002-22-2--2-2ζ16516316716165163ζ16131611ζ16716ζ16716ζ1615169ζ165163    complex lifted from C4○D16
ρ2622-2-200-2i2i00000002-22-2--2-2165163ζ16716ζ165163ζ165163ζ161516916716ζ16716ζ16131611    complex lifted from C4○D16
ρ274-4-440-400400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-44-400000000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ294-44-4000000000002222-22-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

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

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

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

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

Matrix representation of C23.20D8 in GL4(𝔽17) generated by

1000
11600
0010
00016
,
16000
01600
0010
0001
,
1000
0100
00160
00016
,
13800
13400
00100
0005
,
11500
01600
0005
00100
G:=sub<GL(4,GF(17))| [1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[13,13,0,0,8,4,0,0,0,0,10,0,0,0,0,5],[1,0,0,0,15,16,0,0,0,0,0,10,0,0,5,0] >;

C23.20D8 in GAP, Magma, Sage, TeX

C_2^3._{20}D_8
% in TeX

G:=Group("C2^3.20D8");
// GroupNames label

G:=SmallGroup(128,969);
// by ID

G=gap.SmallGroup(128,969);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,456,422,58,1684,438,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of C23.20D8 in TeX

׿
×
𝔽