Copied to
clipboard

G = C24.55D4order 128 = 27

10th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.55D4, C22⋊Q84C4, (C22×Q8)⋊3C4, C22.24C4≀C2, C4.12(C23⋊C4), C23.498(C2×D4), (C22×C4).737D4, C23.31D418C2, C22⋊C8.130C22, C23.53(C22⋊C4), C24.4C4.13C2, (C22×C4).630C23, (C23×C4).208C22, C22⋊Q8.139C22, C2.8(C23.38D4), C22.24(C8.C22), C2.C42.506C22, (C2×C4⋊C4)⋊8C4, C4⋊C4.8(C2×C4), C2.25(C2×C4≀C2), (C2×Q8).7(C2×C4), C2.17(C2×C23⋊C4), (C2×C22⋊Q8).3C2, (C2×C4).1154(C2×D4), (C4×C22⋊C4).10C2, (C2×C4).120(C22×C4), (C22×C4).199(C2×C4), (C2×C4).172(C22⋊C4), C22.184(C2×C22⋊C4), SmallGroup(128,240)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.55D4
C1C2C22C23C22×C4C23×C4C2×C22⋊Q8 — C24.55D4
C1C22C2×C4 — C24.55D4
C1C22C23×C4 — C24.55D4
C1C2C22C22×C4 — C24.55D4

Generators and relations for C24.55D4
 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=bcde3 >

Subgroups: 324 in 147 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×11], C8 [×2], C2×C4 [×4], C2×C4 [×25], Q8 [×4], C23 [×3], C23 [×5], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×2], C22×C4 [×6], C22×C4 [×6], C2×Q8 [×2], C2×Q8 [×3], C24, C2.C42 [×2], C22⋊C8 [×2], C22⋊C8, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C2×M4(2), C23×C4, C22×Q8, C23.31D4 [×4], C4×C22⋊C4, C24.4C4, C2×C22⋊Q8, C24.55D4
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], C4≀C2 [×2], C2×C22⋊C4, C8.C22 [×2], C2×C23⋊C4, C23.38D4, C2×C4≀C2, C24.55D4

Smallest permutation representation of C24.55D4
On 32 points
Generators in S32
(1 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)(17 27)(18 32)(19 29)(20 26)(21 31)(22 28)(23 25)(24 30)
(1 5)(2 30)(3 7)(4 32)(6 26)(8 28)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)(25 29)(27 31)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 17 5 21)(2 24 30 11)(3 14 7 10)(4 13 32 18)(6 20 26 15)(8 9 28 22)(12 25 16 29)(19 27 23 31)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G···4O4P4Q4R4S8A8B8C8D
order1222222224···44···444448888
size1111222242···24···488888888

32 irreducible representations

dim1111111122244
type++++++++-
imageC1C2C2C2C2C4C4C4D4D4C4≀C2C23⋊C4C8.C22
kernelC24.55D4C23.31D4C4×C22⋊C4C24.4C4C2×C22⋊Q8C2×C4⋊C4C22⋊Q8C22×Q8C22×C4C24C22C4C22
# reps1411124231822

Matrix representation of C24.55D4 in GL6(𝔽17)

100000
0160000
0011500
0001600
00161016
00161160
,
1600000
010000
001000
000100
00160160
00160016
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
0160000
400000
001002
001011
0000016
00161016
,
400000
0160000
001000
0011600
00160016
000010

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,16,0,0,0,0,0,1,0,0,0,1,0,0,0,0,2,1,16,16],[4,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;

C24.55D4 in GAP, Magma, Sage, TeX

C_2^4._{55}D_4
% in TeX

G:=Group("C2^4.55D4");
// GroupNames label

G:=SmallGroup(128,240);
// by ID

G=gap.SmallGroup(128,240);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,387,352,1123,1018,248,1971]);
// 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*c*d*e^3>;
// generators/relations

׿
×
𝔽