Copied to
clipboard

G = C24.60D4order 128 = 27

15th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.60D4, C23.10D8, C23.15SD16, C4⋊D48C4, (C22×D4)⋊5C4, C22.12(C2×D8), C4.13(C23⋊C4), C22⋊C850C22, C22.SD163C2, (C22×C4).220D4, C23.509(C2×D4), C23.7Q82C2, C22.30(C2×SD16), C4⋊D4.142C22, C2.C422C22, (C22×C4).641C23, (C23×C4).213C22, C22.25(D4⋊C4), C23.175(C22⋊C4), C2.24(C42⋊C22), (C2×C4⋊C4)⋊11C4, C4⋊C4.19(C2×C4), (C2×C22⋊C8)⋊8C2, (C2×D4).15(C2×C4), (C2×C4⋊D4).5C2, C2.22(C2×C23⋊C4), C2.10(C2×D4⋊C4), (C2×C4).1165(C2×D4), (C2×C4).131(C22×C4), (C22×C4).204(C2×C4), (C2×C4).241(C22⋊C4), C22.195(C2×C22⋊C4), SmallGroup(128,251)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.60D4
C1C2C22C23C22×C4C23×C4C2×C4⋊D4 — C24.60D4
C1C22C2×C4 — C24.60D4
C1C22C23×C4 — C24.60D4
C1C2C22C22×C4 — C24.60D4

Generators and relations for C24.60D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, faf-1=acd, 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 170 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×20], C8 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×12], C23 [×3], C23 [×4], C23 [×10], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], C22×C4 [×6], C22×C4 [×5], C2×D4 [×2], C2×D4 [×11], C24, C24, C2.C42 [×2], C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22×C8, C23×C4, C22×D4, C22×D4, C22.SD16 [×4], C23.7Q8, C2×C22⋊C8, C2×C4⋊D4, C24.60D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C23⋊C4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×C23⋊C4, C2×D4⋊C4, C42⋊C22, C24.60D4

Smallest permutation representation of C24.60D4
On 32 points
Generators in S32
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(1 14)(2 31)(3 16)(4 25)(5 10)(6 27)(7 12)(8 29)(9 19)(11 21)(13 23)(15 17)(18 32)(20 26)(22 28)(24 30)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(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 29 14 8)(2 22 31 28)(3 11 16 21)(4 5 25 10)(6 18 27 32)(7 15 12 17)(9 26 19 20)(13 30 23 24)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E4F4G···4L8A···8H
order12222···2224444444···48···8
size11112···2882222448···84···4

32 irreducible representations

dim11111111222244
type+++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16C23⋊C4C42⋊C22
kernelC24.60D4C22.SD16C23.7Q8C2×C22⋊C8C2×C4⋊D4C2×C4⋊C4C4⋊D4C22×D4C22×C4C24C23C23C4C2
# reps14111242314422

Matrix representation of C24.60D4 in GL6(𝔽17)

100000
010000
00161500
000100
0044016
001313160
,
100000
010000
00161500
000100
000401
0001310
,
1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
3140000
330000
00130150
0040116
00161640
0000130
,
3140000
14140000
00130150
000011
000140
0000130

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,4,13,0,0,15,1,4,13,0,0,0,0,0,16,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1,4,13,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,13,4,16,0,0,0,0,0,16,0,0,0,15,1,4,13,0,0,0,16,0,0],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,15,1,4,13,0,0,0,1,0,0] >;

C24.60D4 in GAP, Magma, Sage, TeX

C_2^4._{60}D_4
% in TeX

G:=Group("C2^4.60D4");
// GroupNames label

G:=SmallGroup(128,251);
// by ID

G=gap.SmallGroup(128,251);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,352,1123,1018,248,1971]);
// 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,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*c*d,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

׿
×
𝔽