Copied to
clipboard

G = C23.(C2×D4)  order 128 = 27

6th non-split extension by C23 of C2×D4 acting via C2×D4/C2=D4

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

Aliases: (C22×Q8)⋊8C4, C23.6(C2×D4), C42⋊C28C4, (C2×D4).128D4, (C2×Q8).116D4, (C22×C4).93D4, C4.17(C23⋊C4), (C2×D4).17C23, C23.D46C2, C22.D43C4, C23⋊C4.10C22, C23.56(C22×C4), C23.21(C22⋊C4), C4.D4.11C22, C23.C23.9C2, C22.D4.2C22, C23.38C23.6C2, M4(2).8C22.10C2, (C2×C4).6(C2×D4), C22⋊C4.3(C2×C4), C2.35(C2×C23⋊C4), (C2×D4).126(C2×C4), (C22×C4).30(C2×C4), (C2×C4○D4).73C22, C22.59(C2×C22⋊C4), (C2×C4).145(C22⋊C4), SmallGroup(128,855)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.(C2×D4)
C1C2C22C23C2×D4C2×C4○D4C23.38C23 — C23.(C2×D4)
C1C2C22C23 — C23.(C2×D4)
C1C2C2×C4C2×C4○D4 — C23.(C2×D4)
C1C2C22C2×D4 — C23.(C2×D4)

Generators and relations for C23.(C2×D4)
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e4=c, f2=ca=dad-1=ac, ab=ba, eae-1=abc, af=fa, ebe-1=fbf-1=bc=cb, ede-1=bd=db, cd=dc, ce=ec, cf=fc, fdf-1=bcd, fef-1=ace3 >

Subgroups: 268 in 117 conjugacy classes, 42 normal (26 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×9], C22, C22 [×6], C8 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×3], Q8 [×5], C23 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8, M4(2) [×3], C22×C4 [×3], C22×C4 [×2], C2×D4 [×3], C2×Q8, C2×Q8 [×4], C4○D4 [×2], C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.10D4, C42⋊C2, C42⋊C2, C22⋊Q8 [×2], C22.D4 [×4], C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C23.D4 [×4], C23.C23, M4(2).8C22, C23.38C23, C23.(C2×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C23⋊C4, C23.(C2×D4)

Character table of C23.(C2×D4)

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11244422444888888888888
ρ111111111111111111111111    trivial
ρ211111-1-1-1-11-1-1-1-111-11111-1-1    linear of order 2
ρ311111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ411111-1-1-1-11-1111-1-11-1-111-1-1    linear of order 2
ρ511111-1-1-1-11-11-1-11-111-1-1-111    linear of order 2
ρ611111111111-1111-1-11-1-1-1-1-1    linear of order 2
ρ711111-1-1-1-11-1-111-11-1-11-1-111    linear of order 2
ρ8111111111111-1-1-111-11-1-1-1-1    linear of order 2
ρ9111-111-1-1-1-11-i-11-1-ii1i-iii-i    linear of order 4
ρ10111-11-1111-1-1i1-1-1-i-i1i-ii-ii    linear of order 4
ρ11111-111-1-1-1-11i1-11i-i-1-i-iii-i    linear of order 4
ρ12111-11-1111-1-1-i-111ii-1-i-ii-ii    linear of order 4
ρ13111-11-1111-1-1-i1-1-1ii1-ii-ii-i    linear of order 4
ρ14111-111-1-1-1-11i-11-1i-i1-ii-i-ii    linear of order 4
ρ15111-11-1111-1-1i-111-i-i-1ii-ii-i    linear of order 4
ρ16111-111-1-1-1-11-i1-11-ii-1ii-i-ii    linear of order 4
ρ172222-2-222-2-22000000000000    orthogonal lifted from D4
ρ18222-2-2222-22-2000000000000    orthogonal lifted from D4
ρ192222-22-2-22-2-2000000000000    orthogonal lifted from D4
ρ20222-2-2-2-2-2222000000000000    orthogonal lifted from D4
ρ2144-40004-4000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000-44000000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C23.(C2×D4)
On 32 points
Generators in S32
(2 26)(3 7)(4 32)(6 30)(8 28)(9 13)(10 22)(12 20)(14 18)(16 24)(17 21)(27 31)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 21 5 17)(2 10 6 14)(3 19 7 23)(4 16 8 12)(9 25 13 29)(11 31 15 27)(18 30 22 26)(20 28 24 32)
(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)
(1 23 5 19)(2 10 30 18)(3 13)(4 20 28 16)(6 14 26 22)(7 9)(8 24 32 12)(11 29 15 25)(17 27)(21 31)

G:=sub<Sym(32)| (2,26)(3,7)(4,32)(6,30)(8,28)(9,13)(10,22)(12,20)(14,18)(16,24)(17,21)(27,31), (1,29)(2,26)(3,31)(4,28)(5,25)(6,30)(7,27)(8,32)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,21,5,17)(2,10,6,14)(3,19,7,23)(4,16,8,12)(9,25,13,29)(11,31,15,27)(18,30,22,26)(20,28,24,32), (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), (1,23,5,19)(2,10,30,18)(3,13)(4,20,28,16)(6,14,26,22)(7,9)(8,24,32,12)(11,29,15,25)(17,27)(21,31)>;

G:=Group( (2,26)(3,7)(4,32)(6,30)(8,28)(9,13)(10,22)(12,20)(14,18)(16,24)(17,21)(27,31), (1,29)(2,26)(3,31)(4,28)(5,25)(6,30)(7,27)(8,32)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,21,5,17)(2,10,6,14)(3,19,7,23)(4,16,8,12)(9,25,13,29)(11,31,15,27)(18,30,22,26)(20,28,24,32), (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), (1,23,5,19)(2,10,30,18)(3,13)(4,20,28,16)(6,14,26,22)(7,9)(8,24,32,12)(11,29,15,25)(17,27)(21,31) );

G=PermutationGroup([(2,26),(3,7),(4,32),(6,30),(8,28),(9,13),(10,22),(12,20),(14,18),(16,24),(17,21),(27,31)], [(1,29),(2,26),(3,31),(4,28),(5,25),(6,30),(7,27),(8,32),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(1,21,5,17),(2,10,6,14),(3,19,7,23),(4,16,8,12),(9,25,13,29),(11,31,15,27),(18,30,22,26),(20,28,24,32)], [(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)], [(1,23,5,19),(2,10,30,18),(3,13),(4,20,28,16),(6,14,26,22),(7,9),(8,24,32,12),(11,29,15,25),(17,27),(21,31)])

Matrix representation of C23.(C2×D4) in GL8(𝔽17)

10000000
01000000
001600000
000160000
00001000
00000100
000000160
000000016
,
1615000000
01000000
0016150000
00010000
01000100
016001000
00010001
000160010
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
0016000150
00000011
1600015000
00001100
00100010
00161600160
10001000
16160016000
,
1300013444
400041300
001301313134
004000413
0150152151515
13150215222
0201522215
01513151515152
,
1600015000
100011600
001600002
001000116
10001000
010016000
001100016
00000001

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

C23.(C2×D4) in GAP, Magma, Sage, TeX

C_2^3.(C_2\times D_4)
% in TeX

G:=Group("C2^3.(C2xD4)");
// GroupNames label

G:=SmallGroup(128,855);
// by ID

G=gap.SmallGroup(128,855);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,851,375,4037]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=e^4=c,f^2=c*a=d*a*d^-1=a*c,a*b=b*a,e*a*e^-1=a*b*c,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=b*c*d,f*e*f^-1=a*c*e^3>;
// generators/relations

Export

Character table of C23.(C2×D4) in TeX

׿
×
𝔽