Copied to
clipboard

G = C24.54D4  order 128 = 27

9th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.54D4, C4⋊D4⋊4C4, (C22×D4)⋊3C4, C22.23C4≀C2, C4.11(C23⋊C4), C22⋊C8⋊41C22, (C22×C4).736D4, C23.497(C2×D4), C24.4C4⋊20C2, C22.SD16⋊19C2, C4⋊D4.134C22, C22.35(C8⋊C22), C23.52(C22⋊C4), (C22×C4).629C23, (C23×C4).207C22, C2.C42⋊57C22, C2.8(C23.37D4), (C2×C4⋊C4)⋊7C4, C4⋊C4.7(C2×C4), C2.24(C2×C4≀C2), (C2×D4).7(C2×C4), (C4×C22⋊C4)⋊21C2, (C2×C4⋊D4).3C2, C2.16(C2×C23⋊C4), (C2×C4).1153(C2×D4), (C2×C4).119(C22×C4), (C22×C4).198(C2×C4), (C2×C4).171(C22⋊C4), C22.183(C2×C22⋊C4), SmallGroup(128,239)

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

Derived series
C1 — C2×C4 — C24.54D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4⋊D4 — C24.54D4
Lower central
C1 — C22 — C2×C4 — C24.54D4
Upper central
C1 — C22 — C23×C4 — C24.54D4
Jennings
C1 — C2 — C22 — C22×C4 — C24.54D4

Generators and relations for C24.54D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, faf-1=ac=ca, eae-1=ad=da, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 452 in 173 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C22⋊C8, C22⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C2×M4(2), C23×C4, C22×D4, C22×D4, C22.SD16, C4×C22⋊C4, C24.4C4, C2×C4⋊D4, C24.54D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C4≀C2, C2×C22⋊C4, C8⋊C22, C2×C23⋊C4, C23.37D4, C2×C4≀C2, C24.54D4

Smallest permutation representation of C24.54D4
►On 32 points
Generators in S32
(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)(17 27)(18 32)(19 29)(20 26)(21 31)(22 28)(23 25)(24 30)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4F4G···4O4P4Q8A8B8C8D
order122222222224···44···4448888
size111122224882···24···4888888

32 irreducible representations

dim1111111122244
type+++++++++
imageC1C2C2C2C2C4C4C4D4D4C4≀C2C23⋊C4C8⋊C22
kernelC24.54D4C22.SD16C4×C22⋊C4C24.4C4C2×C4⋊D4C2×C4⋊C4C4⋊D4C22×D4C22×C4C24C22C4C22
# reps1411124231822

Matrix representation of C24.54D4 ►in GL6(𝔽17)

1150000
0160000
0016000
0001600
000010
000001
,
1150000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
16140000
1610000
000010
0000016
0001600
0016000
,
130000
040000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,14,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C24.54D4 in GAP, Magma, Sage, TeX 

C_2^4._{54}D_4
 
% in TeX
 
G:=Group("C2^4.54D4");
 
// GroupNames label
 
G:=SmallGroup(128,239);
 
// by ID
 
G=gap.SmallGroup(128,239);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,352,1123,1018,248,1971]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,f*a*f^-1=a*c=c*a,e*a*e^-1=a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*d*e^3>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁