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G = C24.121D4order 128 = 27

76th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.121D4, C4.32+ 1+4, C88D41C2, C22⋊D85C2, D4⋊D41C2, C87D417C2, (C2×D8)⋊3C22, C2.D84C22, C4.Q832C22, C22⋊SD1629C2, C4⋊D457C22, C4⋊C4.127C23, C22⋊C860C22, (C2×C4).386C24, (C2×C8).320C23, (C22×C8)⋊16C22, Q8⋊C41C22, C23.271(C2×D4), (C22×C4).484D4, C22⋊Q869C22, D4⋊C443C22, C22.19C249C2, (C2×SD16)⋊39C22, (C2×D4).139C23, C22.31(C4○D8), C23.48D44C2, C23.19D41C2, C22.2(C8⋊C22), (C2×Q8).126C23, C42⋊C216C22, C23.46D429C2, C2.67(C233D4), (C23×C4).566C22, C22.646(C22×D4), (C22×C4).1064C23, (C22×D4).381C22, C2.38(C2×C4○D8), (C2×C4⋊D4)⋊50C2, (C2×C22⋊C8)⋊28C2, (C2×C4).704(C2×D4), (C2×C4○D4)⋊8C22, C2.49(C2×C8⋊C22), (C2×C4⋊C4).636C22, SmallGroup(128,1920)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.121D4
C1C2C4C2×C4C22×C4C22×D4C2×C4⋊D4 — C24.121D4
C1C2C2×C4 — C24.121D4
C1C22C23×C4 — C24.121D4
C1C2C2C2×C4 — C24.121D4

Generators and relations for C24.121D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, faf=acd, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Subgroups: 556 in 236 conjugacy classes, 88 normal (42 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×24], C8 [×4], C2×C4 [×4], C2×C4 [×18], D4 [×22], Q8 [×2], C23 [×3], C23 [×14], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×14], C2×Q8, C4○D4 [×2], C24, C24, C22⋊C8 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8 [×2], C2×D8 [×2], C2×SD16 [×2], C23×C4, C22×D4, C22×D4, C2×C4○D4, C2×C22⋊C8, C22⋊D8, D4⋊D4 [×2], C22⋊SD16, C88D4 [×2], C87D4 [×2], C23.46D4, C23.19D4 [×2], C23.48D4, C2×C4⋊D4, C22.19C24, C24.121D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8⋊C22 [×2], C22×D4, 2+ 1+4 [×2], C233D4, C2×C4○D8, C2×C8⋊C22, C24.121D4

Smallest permutation representation of C24.121D4
On 32 points
Generators in S32
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(2 30)(4 32)(6 26)(8 28)(9 19)(11 21)(13 23)(15 17)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(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 2)(3 8)(4 7)(5 6)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(25 26)(27 32)(28 31)(29 30)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A···4F4G4H···4L8A···8H
order1222222222224···444···48···8
size1111222248882···248···84···4

32 irreducible representations

dim11111111111122244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D82+ 1+4C8⋊C22
kernelC24.121D4C2×C22⋊C8C22⋊D8D4⋊D4C22⋊SD16C88D4C87D4C23.46D4C23.19D4C23.48D4C2×C4⋊D4C22.19C24C22×C4C24C22C4C22
# reps11121221211131822

Matrix representation of C24.121D4 in GL6(𝔽17)

490000
4130000
001200
0001600
002212
001515016
,
1600000
0160000
001000
000100
0000160
001515016
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
060000
1460000
000010
001101
001000
001601616
,
1160000
1460000
000010
0016161616
001000
000001

G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,9,13,0,0,0,0,0,0,1,0,2,15,0,0,2,16,2,15,0,0,0,0,1,0,0,0,0,0,2,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,15,0,0,0,1,0,15,0,0,0,0,16,0,0,0,0,0,0,16],[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],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,1,1,16,0,0,0,1,0,0,0,0,1,0,0,16,0,0,0,1,0,16],[11,14,0,0,0,0,6,6,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,0,0,16,0,1] >;

C24.121D4 in GAP, Magma, Sage, TeX

C_2^4._{121}D_4
% in TeX

G:=Group("C2^4.121D4");
// GroupNames label

G:=SmallGroup(128,1920);
// by ID

G=gap.SmallGroup(128,1920);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,4037,1027,124]);
// 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,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a*c*d,e*b*e^-1=f*b*f=b*c=c*b,b*d=d*b,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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