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G = C24.130D4  order 128 = 27

85th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.130D4, C4.122+ (1+4), C8⋊2D4⋊11C2, C8⋊D4⋊15C2, D4⋊D4⋊22C2, C8.D4⋊11C2, (C2×D8)⋊24C22, (C2×C8).60C23, C4.Q8⋊19C22, C2.D8⋊30C22, D4.7D4⋊23C2, C4⋊C4.136C23, (C2×C4).395C24, (C2×Q16)⋊24C22, (C22×C4).492D4, C23.279(C2×D4), D4⋊C4⋊28C22, C24.4C4⋊14C2, Q8⋊C4⋊31C22, (C2×SD16)⋊21C22, (C2×D4).146C23, C22⋊C8.41C22, (C2×Q8).134C23, C22.19C24⋊12C2, C23.20D4⋊23C2, C23.19D4⋊23C2, C4⋊D4.185C22, C2.76(C23⋊3D4), (C23×C4).575C22, C22.655(C22×D4), C22⋊Q8.190C22, C2.53(D8⋊C22), (C22×C4).1073C23, (C2×M4(2)).82C22, C42⋊C2.153C22, (C2×C4).533(C2×D4), (C2×C4○D4).164C22, SmallGroup(128,1929)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C24.130D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.19C24 — C24.130D4
Lower central
C1 — C2 — C2×C4 — C24.130D4
Upper central
C1 — C22 — C23×C4 — C24.130D4
Jennings
C1 — C2 — C2 — C2×C4 — C24.130D4

Subgroups: 452 in 209 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×19], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×19], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×5], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8, SD16 [×2], Q16, C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×4], C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×2], C2×M4(2) [×2], C2×D8, C2×SD16 [×2], C2×Q16, C23×C4, C2×C4○D4 [×2], C24.4C4, D4⋊D4 [×2], D4.7D4 [×2], C8⋊D4 [×2], C8⋊2D4, C8.D4, C23.19D4 [×2], C23.20D4 [×2], C22.19C24 [×2], C24.130D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×2], C23⋊3D4, D8⋊C22 [×2], C24.130D4

Generators and relations
 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=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Smallest permutation representation
►On 32 points
Generators in S32
(2 6)(4 8)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)(25 29)(27 31)

(1 5)(2 27)(3 7)(4 29)(6 31)(8 25)(9 24)(11 18)(13 20)(15 22)(26 30)(28 32)

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

Matrix representation ►G ⊆ GL8(𝔽17)

10000000
016000000
1601600000
01010000
00001000
000001600
00000010
000000016
,
10000000
01000000
1601600000
0160160000
000016000
00000100
00000010
000000016
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
0160150000
1601500000
00010000
00100000
00000001
00000010
000001300
00004000
,
1601500000
0160150000
00100000
00010000
00000010
00000001
00001000
00000100

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

Character table of C24.130D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11114448822222248888888888
ρ111111111111111111111111111    trivial
ρ211111-1-1-1-11-1-11-1-1111-11-111-11-1    linear of order 2
ρ311111-1-1111-1-11-1-11-1-11-11-11-11-1    linear of order 2
ρ41111111-1-11111111-1-1-1-1-1-11111    linear of order 2
ρ511111-1-1-111-1-11-1-1111-1-11-1-11-11    linear of order 2
ρ611111111-11111111111-1-1-1-1-1-1-1    linear of order 2
ρ71111111-111111111-1-1-1111-1-1-1-1    linear of order 2
ρ811111-1-11-11-1-11-1-11-1-111-11-11-11    linear of order 2
ρ91111-11-1-1-11-1111-1-1-111-111-1-111    linear of order 2
ρ101111-1-111111-11-11-1-11-1-1-11-111-1    linear of order 2
ρ111111-1-11-1-111-11-11-11-1111-1-111-1    linear of order 2
ρ121111-11-1111-1111-1-11-1-11-1-1-1-111    linear of order 2
ρ131111-1-111-111-11-11-1-11-111-11-1-11    linear of order 2
ρ141111-11-1-111-1111-1-1-1111-1-111-1-1    linear of order 2
ρ151111-11-11-11-1111-1-11-1-1-11111-1-1    linear of order 2
ρ161111-1-11-1111-11-11-11-11-1-111-1-11    linear of order 2
ρ172222-22-200-22-2-2-2220000000000    orthogonal lifted from D4
ρ182222-2-2200-2-22-22-220000000000    orthogonal lifted from D4
ρ1922222-2-200-222-222-20000000000    orthogonal lifted from D4
ρ20222222200-2-2-2-2-2-2-20000000000    orthogonal lifted from D4
ρ214-44-400000400-40000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-400000-40040000000000000    orthogonal lifted from 2+ (1+4)
ρ2344-4-40000004i0004i00000000000    complex lifted from D8⋊C22
ρ244-4-4400000004i04i000000000000    complex lifted from D8⋊C22
ρ2544-4-40000004i0004i00000000000    complex lifted from D8⋊C22
ρ264-4-4400000004i04i000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_2^4._{130}D_4
 
% in TeX
 
G:=Group("C2^4.130D4");
 
// GroupNames label
 
G:=SmallGroup(128,1929);
 
// by ID
 
G=gap.SmallGroup(128,1929);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,891,675,248,4037,1027,124]);
 
// 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=b*c=c*b,b*d=d*b,f*b*f=b*c*d,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
 

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