Copied to
clipboard

G = C24.125D4order 128 = 27

80th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.125D4, C4.72+ 1+4, C82D49C2, C22⋊D817C2, (C2×D8)⋊22C22, (C2×C8).55C23, C4.Q815C22, C4⋊D458C22, C4⋊C4.131C23, C22⋊C813C22, (C2×C4).390C24, (C22×C4).487D4, C23.274(C2×D4), D4⋊C424C22, C24.4C410C2, (C2×D4).142C23, C23.46D48C2, (C22×D4)⋊24C22, C2.71(C233D4), C22.49(C8⋊C22), (C2×M4(2))⋊13C22, (C23×C4).570C22, C22.650(C22×D4), (C22×C4).1068C23, (C2×C4⋊D4)⋊51C2, (C2×C4⋊C4)⋊52C22, (C2×C4).528(C2×D4), C2.50(C2×C8⋊C22), SmallGroup(128,1924)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.125D4
C1C2C4C2×C4C22×C4C22×D4C2×C4⋊D4 — C24.125D4
C1C2C2×C4 — C24.125D4
C1C22C23×C4 — C24.125D4
C1C2C2C2×C4 — C24.125D4

Generators and relations for C24.125D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, faf=ac=ca, eae-1=ad=da, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Subgroups: 660 in 257 conjugacy classes, 88 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×9], C4 [×2], C4 [×7], C22, C22 [×4], C22 [×31], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×28], C23, C23 [×2], C23 [×21], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×2], D8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C2×D4 [×4], C2×D4 [×22], C24, C24 [×2], C22⋊C8 [×4], D4⋊C4 [×8], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C4⋊D4 [×4], C2×M4(2) [×2], C2×D8 [×4], C23×C4, C22×D4 [×2], C22×D4 [×2], C24.4C4, C22⋊D8 [×4], C82D4 [×4], C23.46D4 [×4], C2×C4⋊D4 [×2], C24.125D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×4], C22×D4, 2+ 1+4 [×2], C233D4, C2×C8⋊C22 [×2], C24.125D4

Character table of C24.125D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112222488882244488888888
ρ111111111111111111111111111    trivial
ρ211111-1-11-11-1-1-1111-1-1-1111-11-11    linear of order 2
ρ31111-1-1-1-11-1-11111-1-11-11-1111-1-1    linear of order 2
ρ41111-111-1-1-11-1-111-11-111-11-111-1    linear of order 2
ρ5111111111-111-111111-1-111-1-1-1-1    linear of order 2
ρ611111-1-11-1-1-1-11111-1-11-1111-11-1    linear of order 2
ρ71111-1-1-1-111-11-111-1-111-1-11-1-111    linear of order 2
ρ81111-111-1-111-1111-11-1-1-1-111-1-11    linear of order 2
ρ91111-1-1-1-1111-1-111-1-111-11-111-1-1    linear of order 2
ρ101111-111-1-11-11111-11-1-1-11-1-111-1    linear of order 2
ρ11111111111-1-1-1-111111-1-1-1-11111    linear of order 2
ρ1211111-1-11-1-1111111-1-11-1-1-1-11-11    linear of order 2
ρ131111-1-1-1-11-11-1111-1-11-111-1-1-111    linear of order 2
ρ141111-111-1-1-1-11-111-11-1111-11-1-11    linear of order 2
ρ151111111111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ1611111-1-11-1111-1111-1-1-11-1-11-11-1    linear of order 2
ρ172222222220000-2-2-2-2-200000000    orthogonal lifted from D4
ρ182222-2-2-2-220000-2-222-200000000    orthogonal lifted from D4
ρ192222-222-2-20000-2-22-2200000000    orthogonal lifted from D4
ρ2022222-2-22-20000-2-2-22200000000    orthogonal lifted from D4
ρ214-4-4404-40000000000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4-4004000000000000000000    orthogonal lifted from C8⋊C22
ρ234-4-440-440000000000000000000    orthogonal lifted from C8⋊C22
ρ244-44-4000000000-4400000000000    orthogonal lifted from 2+ 1+4
ρ2544-4-4400-4000000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-40000000004-400000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of C24.125D4
On 32 points
Generators in S32
(1 18)(2 23)(3 20)(4 17)(5 22)(6 19)(7 24)(8 21)(9 27)(10 32)(11 29)(12 26)(13 31)(14 28)(15 25)(16 30)
(2 30)(4 32)(6 26)(8 28)(10 17)(12 19)(14 21)(16 23)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 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 2)(3 8)(4 7)(5 6)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 26)(27 32)(28 31)(29 30)

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

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

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

Matrix representation of C24.125D4 in GL8(𝔽17)

115000000
016000000
001620000
00010000
00000010
00000001
00001000
00000100
,
10000000
01000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
001600000
001610000
10000000
116000000
00000006
0000001411
000001100
00003600
,
00100000
00010000
10000000
01000000
00000066
0000001411
00006600
0000141100

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

C24.125D4 in GAP, Magma, Sage, TeX

C_2^4._{125}D_4
% in TeX

G:=Group("C2^4.125D4");
// GroupNames label

G:=SmallGroup(128,1924);
// by ID

G=gap.SmallGroup(128,1924);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,891,675,4037,1027,124]);
// Polycyclic

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

Export

Character table of C24.125D4 in TeX

׿
×
𝔽