Copied to
clipboard

G = C24.98D4order 128 = 27

53rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.98D4, C4.7(C23×C4), C4⋊C4.343C23, (C2×C8).392C23, (C2×C4).177C24, (C22×C8)⋊48C22, D4.19(C22×C4), (C22×C4).781D4, C23.377(C2×D4), C4.142(C22×D4), Q8.19(C22×C4), D4⋊C485C22, Q8⋊C488C22, (C2×D4).361C23, C4(C23.37D4), C4(C23.36D4), C4(C23.38D4), (C2×Q8).334C23, C42⋊C275C22, C23.36D447C2, C23.24D434C2, C2.1(D8⋊C22), C23.38D438C2, C23.37D438C2, C23.87(C22⋊C4), (C22×M4(2))⋊20C2, (C2×M4(2))⋊70C22, (C23×C4).515C22, (C22×C4).901C23, C22.127(C22×D4), (C22×D4).554C22, (C22×Q8).458C22, (C2×C4○D4)⋊20C4, C4○D414(C2×C4), (C2×D4)⋊49(C2×C4), (C2×Q8)⋊40(C2×C4), (C2×C4).445(C2×D4), C4.75(C2×C22⋊C4), (C2×C4⋊C4)⋊115C22, (C2×C42⋊C2)⋊42C2, (C2×C4).245(C22×C4), (C22×C4).325(C2×C4), (C22×C4○D4).20C2, C22.22(C2×C22⋊C4), C2.39(C22×C22⋊C4), (C2×C4).159(C22⋊C4), (C2×C4○D4).274C22, (C2×C4)(C23.36D4), (C2×C4)(C23.38D4), SmallGroup(128,1628)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C24.98D4
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C24.98D4
C1C2C4 — C24.98D4
C1C2×C4C23×C4 — C24.98D4
C1C2C2C2×C4 — C24.98D4

Generators and relations for C24.98D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 668 in 386 conjugacy classes, 172 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×26], C8 [×4], C2×C4 [×4], C2×C4 [×24], C2×C4 [×30], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23 [×3], C23 [×4], C23 [×12], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×6], C22×C4 [×8], C22×C4 [×15], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×16], C4○D4 [×24], C24, C24, D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C23.24D4 [×4], C23.36D4 [×4], C23.37D4 [×2], C23.38D4 [×2], C2×C42⋊C2, C22×M4(2), C22×C4○D4, C24.98D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, D8⋊C22 [×2], C24.98D4

Smallest permutation representation of C24.98D4
On 32 points
Generators in S32
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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 13 26 22)(2 21 27 12)(3 11 28 20)(4 19 29 10)(5 9 30 18)(6 17 31 16)(7 15 32 24)(8 23 25 14)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E···4J4K···4V8A···8H
order12222···2222244444···44···48···8
size11112···2444411112···24···44···4

44 irreducible representations

dim111111111224
type++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4D8⋊C22
kernelC24.98D4C23.24D4C23.36D4C23.37D4C23.38D4C2×C42⋊C2C22×M4(2)C22×C4○D4C2×C4○D4C22×C4C24C2
# reps1442211116714

Matrix representation of C24.98D4 in GL6(𝔽17)

1600000
0160000
001000
000100
0010160
00160016
,
1600000
0160000
0016000
001100
0016010
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
1370000
1040000
00160015
000011
00613016
007001
,
1040000
1370000
0010150
000011
0010160
00161610

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,16,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[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],[13,10,0,0,0,0,7,4,0,0,0,0,0,0,16,0,6,7,0,0,0,0,13,0,0,0,0,1,0,0,0,0,15,1,16,1],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,1,16,0,0,0,0,0,16,0,0,15,1,16,1,0,0,0,1,0,0] >;

C24.98D4 in GAP, Magma, Sage, TeX

C_2^4._{98}D_4
% in TeX

G:=Group("C2^4.98D4");
// GroupNames label

G:=SmallGroup(128,1628);
// by ID

G=gap.SmallGroup(128,1628);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,248,2804,1411,172]);
// Polycyclic

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

׿
×
𝔽