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G = C24.123D4order 128 = 27

78th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.123D4, C4.52+ 1+4, C88D42C2, C2.D86C22, Q8⋊D429C2, D4.7D41C2, C22⋊Q165C2, (C2×Q16)⋊3C22, C4.Q833C22, C8.18D417C2, D4⋊C42C22, C4⋊C4.129C23, (C2×C4).388C24, (C2×C8).152C23, Q8⋊C43C22, C23.272(C2×D4), (C22×C4).486D4, C22.D85C2, (C2×SD16)⋊40C22, (C2×D4).140C23, C22.32(C4○D8), C23.20D41C2, (C2×Q8).128C23, C4⋊D4.181C22, C23.47D429C2, C2.69(C233D4), C22⋊C8.217C22, (C23×C4).568C22, (C22×C8).186C22, C22.648(C22×D4), C22.2(C8.C22), C22⋊Q8.186C22, (C22×C4).1066C23, C22.19C24.20C2, (C22×Q8).314C22, C42⋊C2.150C22, C2.39(C2×C4○D8), (C2×C22⋊C8)⋊29C2, (C2×C4).705(C2×D4), (C2×C22⋊Q8)⋊58C2, C2.49(C2×C8.C22), (C2×C4⋊C4).638C22, (C2×C4○D4).161C22, SmallGroup(128,1922)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.123D4
C1C2C4C2×C4C22×C4C22×Q8C2×C22⋊Q8 — C24.123D4
C1C2C2×C4 — C24.123D4
C1C22C23×C4 — C24.123D4
C1C2C2C2×C4 — C24.123D4

Generators and relations for C24.123D4
 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=e3 >

Subgroups: 428 in 210 conjugacy classes, 88 normal (42 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×4], C22 [×14], C8 [×4], C2×C4 [×4], C2×C4 [×22], D4 [×8], Q8 [×8], C23 [×3], C23 [×6], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4 [×6], C22×C4 [×5], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8 [×2], C2×Q8 [×3], C4○D4 [×2], C24, C22⋊C8 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4, C22×C8 [×2], C2×SD16 [×2], C2×Q16 [×2], C23×C4, C22×Q8, C2×C4○D4, C2×C22⋊C8, Q8⋊D4, C22⋊Q16, D4.7D4 [×2], C88D4 [×2], C8.18D4 [×2], C22.D8, C23.47D4, C23.20D4 [×2], C2×C22⋊Q8, C22.19C24, C24.123D4
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, C24.123D4

Smallest permutation representation of C24.123D4
On 32 points
Generators in S32
(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 5)(2 32)(3 7)(4 26)(6 28)(8 30)(9 13)(10 24)(11 15)(12 18)(14 20)(16 22)(17 21)(19 23)(25 29)(27 31)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(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 16)(2 11)(3 14)(4 9)(5 12)(6 15)(7 10)(8 13)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4F4G4H···4N8A···8H
order12222222224···444···48···8
size11112222482···248···84···4

32 irreducible representations

dim11111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D82+ 1+4C8.C22
kernelC24.123D4C2×C22⋊C8Q8⋊D4C22⋊Q16D4.7D4C88D4C8.18D4C22.D8C23.47D4C23.20D4C2×C22⋊Q8C22.19C24C22×C4C24C22C4C22
# reps11112221121131822

Matrix representation of C24.123D4 in GL6(𝔽17)

100000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0014810
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
1500000
090000
001511715
0000012
000020
0010700
,
090000
200000
000001
007417
001621316
001000

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,14,0,0,0,0,16,8,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[15,0,0,0,0,0,0,9,0,0,0,0,0,0,15,0,0,10,0,0,11,0,0,7,0,0,7,0,2,0,0,0,15,12,0,0],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,0,7,16,1,0,0,0,4,2,0,0,0,0,1,13,0,0,0,1,7,16,0] >;

C24.123D4 in GAP, Magma, Sage, TeX

C_2^4._{123}D_4
% in TeX

G:=Group("C2^4.123D4");
// GroupNames label

G:=SmallGroup(128,1922);
// by ID

G=gap.SmallGroup(128,1922);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,352,675,4037,1027,124]);
// 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=e^3>;
// generators/relations

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