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G = C24.118D4order 128 = 27

73rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.118D4, (C2×C8).35C23, C2.D821C22, C4.Q810C22, C4⋊C4.393C23, (C2×C4).293C24, C23.244(C2×D4), (C22×C4).444D4, (C2×Q8).69C23, Q8⋊C422C22, C23.47D42C2, C22⋊C8.16C22, C24.4C4.2C2, M4(2)⋊C425C2, C23.38D410C2, C23.48D412C2, C23.20D413C2, (C23×C4).563C22, C22.553(C22×D4), C22⋊Q8.162C22, C2.24(D8⋊C22), (C22×C4).1009C23, C22.35(C8.C22), (C2×M4(2)).75C22, (C22×Q8).292C22, C42⋊C2.318C22, C4.100(C22.D4), C22.65(C22.D4), C4.103(C2×C4○D4), (C2×C4).488(C2×D4), C2.29(C2×C8.C22), (C2×C22⋊Q8).57C2, (C2×C4).488(C4○D4), (C2×C4⋊C4).929C22, (C2×C42⋊C2).60C2, C2.58(C2×C22.D4), SmallGroup(128,1827)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.118D4
C1C2C4C2×C4C22×C4C42⋊C2C2×C42⋊C2 — C24.118D4
C1C2C2×C4 — C24.118D4
C1C22C23×C4 — C24.118D4
C1C2C2C2×C4 — C24.118D4

Generators and relations for C24.118D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 364 in 205 conjugacy classes, 94 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×11], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×26], Q8 [×6], C23 [×3], C23 [×5], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×8], C2×Q8 [×2], C2×Q8 [×5], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C42⋊C2 [×4], C42⋊C2 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C2×M4(2) [×2], C23×C4, C22×Q8, C24.4C4, C23.38D4 [×2], M4(2)⋊C4 [×2], C23.47D4 [×2], C23.48D4 [×2], C23.20D4 [×4], C2×C42⋊C2, C2×C22⋊Q8, C24.118D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C2×C22.D4, C2×C8.C22, D8⋊C22, C24.118D4

Smallest permutation representation of C24.118D4
On 32 points
Generators in S32
(2 32)(4 26)(6 28)(8 30)(9 22)(11 24)(13 18)(15 20)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(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 22 5 18)(2 12 6 16)(3 20 7 24)(4 10 8 14)(9 27 13 31)(11 25 15 29)(17 28 21 32)(19 26 23 30)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111122244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4C4○D4C8.C22D8⋊C22
kernelC24.118D4C24.4C4C23.38D4M4(2)⋊C4C23.47D4C23.48D4C23.20D4C2×C42⋊C2C2×C22⋊Q8C22×C4C24C2×C4C22C2
# reps11222241131822

Matrix representation of C24.118D4 in GL6(𝔽17)

100000
0160000
001080
000102
0000160
0000016
,
100000
010000
0016000
000100
0000160
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
040000
1300000
0002010
0090110
000808
0080150
,
0160000
1600000
0090100
0001506
002080
000202

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,8,0,16,0,0,0,0,2,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[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],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,9,0,8,0,0,2,0,8,0,0,0,0,11,0,15,0,0,10,0,8,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,9,0,2,0,0,0,0,15,0,2,0,0,10,0,8,0,0,0,0,6,0,2] >;

C24.118D4 in GAP, Magma, Sage, TeX

C_2^4._{118}D_4
% in TeX

G:=Group("C2^4.118D4");
// GroupNames label

G:=SmallGroup(128,1827);
// by ID

G=gap.SmallGroup(128,1827);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,2019,248,4037,1027,124]);
// Polycyclic

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

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