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G = C24.106D4order 128 = 27

61st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.106D4, C4○D4.52D4, D4.42(C2×D4), C4⋊C4.9C23, Q8.42(C2×D4), C22⋊SD162C2, C22⋊C86C22, (C2×C8).10C23, C4.44(C22×D4), C4.107C22≀C2, D4.7D414C2, (C2×C4).226C24, C24.4C46C2, C22⋊Q1612C2, (C2×Q16)⋊14C22, (C2×SD16)⋊5C22, C23.649(C2×D4), (C22×C4).788D4, C22⋊Q864C22, (C2×Q8).21C23, D4⋊C411C22, Q8⋊C414C22, C22.18C22≀C2, (C2×D4).381C23, C23.36D42C2, C225(C8.C22), (C2×M4(2))⋊3C22, (C22×Q8)⋊13C22, C2.9(D8⋊C22), (C23×C4).546C22, (C22×C4).964C23, C22.486(C22×D4), (C22×D4).565C22, (C2×C4⋊C4)⋊47C22, (C2×C4).453(C2×D4), (C2×C8.C22)⋊7C2, (C2×C22⋊Q8)⋊53C2, C2.44(C2×C22≀C2), C2.11(C2×C8.C22), (C22×C4○D4).24C2, (C2×C4○D4).297C22, SmallGroup(128,1739)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.106D4
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C24.106D4
C1C2C2×C4 — C24.106D4
C1C22C23×C4 — C24.106D4
C1C2C2C2×C4 — C24.106D4

Generators and relations for C24.106D4
 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=e3 >

Subgroups: 716 in 377 conjugacy classes, 110 normal (28 characteristic)
C1, C2 [×3], C2 [×9], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×27], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×40], D4 [×4], D4 [×22], Q8 [×4], Q8 [×12], C23 [×3], C23 [×15], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4 [×6], C22×C4 [×19], C2×D4 [×2], C2×D4 [×17], C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×8], C4○D4 [×28], C24, C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×4], C2×Q16 [×4], C8.C22 [×8], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8 [×2], C2×C4○D4 [×4], C2×C4○D4 [×10], C24.4C4, C23.36D4 [×2], C22⋊SD16 [×2], C22⋊Q16 [×2], D4.7D4 [×4], C2×C22⋊Q8, C2×C8.C22 [×2], C22×C4○D4, C24.106D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C8.C22 [×2], C22×D4 [×3], C2×C22≀C2, C2×C8.C22, D8⋊C22, C24.106D4

Smallest permutation representation of C24.106D4
On 32 points
Generators in S32
(2 30)(4 32)(6 26)(8 28)(9 22)(11 24)(13 18)(15 20)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
(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 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14)(17 26 21 30)(18 29 22 25)(19 32 23 28)(20 27 24 31)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2L4A···4F4G···4K4L4M4N4O8A8B8C8D
order122222222···24···44···444448888
size111122224···42···24···488888888

32 irreducible representations

dim11111111122244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4D4C8.C22D8⋊C22
kernelC24.106D4C24.4C4C23.36D4C22⋊SD16C22⋊Q16D4.7D4C2×C22⋊Q8C2×C8.C22C22×C4○D4C22×C4C4○D4C24C22C2
# reps11222412138122

Matrix representation of C24.106D4 in GL6(𝔽17)

100000
6160000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1120000
760000
00001212
0000125
0012500
005500
,
1120000
860000
0012500
005500
00001212
0000125

G:=sub<GL(6,GF(17))| [1,6,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[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],[11,7,0,0,0,0,2,6,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,12,12,0,0,0,0,12,5,0,0],[11,8,0,0,0,0,2,6,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,12,12,0,0,0,0,12,5] >;

C24.106D4 in GAP, Magma, Sage, TeX

C_2^4._{106}D_4
% in TeX

G:=Group("C2^4.106D4");
// GroupNames label

G:=SmallGroup(128,1739);
// by ID

G=gap.SmallGroup(128,1739);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,248,2804,1411,172]);
// 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=e^3>;
// generators/relations

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