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G = C24.96D4order 128 = 27

51st non-split extension by C24 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24.96D4, C25.46C22, C23.383C24, C24.575C23, C22.1862+ 1+4, C23.612(C2×D4), C243C4.10C2, (C23×C4).95C22, C23.8Q862C2, C23.305(C4○D4), C23.34D430C2, C23.11D425C2, C2.10(C233D4), (C22×C4).520C23, C22.263(C22×D4), C2.C4225C22, C2.26(C22.45C24), C22.62(C22.D4), (C2×C4⋊C4)⋊20C22, C22.260(C2×C4○D4), (C22×C22⋊C4).13C2, C2.28(C2×C22.D4), (C2×C22⋊C4).461C22, SmallGroup(128,1215)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.96D4
C1C2C22C23C24C23×C4C22×C22⋊C4 — C24.96D4
C1C23 — C24.96D4
C1C23 — C24.96D4
C1C23 — C24.96D4

Generators and relations for C24.96D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde-1 >

Subgroups: 756 in 350 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×10], C4 [×12], C22, C22 [×14], C22 [×50], C2×C4 [×52], C23, C23 [×14], C23 [×50], C22⋊C4 [×26], C4⋊C4 [×4], C22×C4 [×12], C22×C4 [×20], C24, C24 [×6], C24 [×10], C2.C42 [×12], C2×C22⋊C4 [×14], C2×C22⋊C4 [×8], C2×C4⋊C4 [×4], C23×C4 [×4], C25, C243C4, C23.34D4 [×4], C23.8Q8 [×4], C23.11D4 [×4], C22×C22⋊C4 [×2], C24.96D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×8], C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C2×C22.D4 [×2], C233D4, C22.45C24 [×4], C24.96D4

Smallest permutation representation of C24.96D4
On 32 points
Generators in S32
(2 15)(4 13)(6 30)(8 32)(9 17)(10 23)(11 19)(12 21)(18 28)(20 26)(22 27)(24 25)
(1 31)(2 15)(3 29)(4 13)(5 16)(6 30)(7 14)(8 32)(9 27)(10 23)(11 25)(12 21)(17 22)(18 28)(19 24)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 17)(10 18)(11 19)(12 20)(13 30)(14 31)(15 32)(16 29)(21 26)(22 27)(23 28)(24 25)
(1 31)(2 32)(3 29)(4 30)(5 16)(6 13)(7 14)(8 15)(9 22)(10 23)(11 24)(12 21)(17 27)(18 28)(19 25)(20 26)
(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 17 31 27)(2 21 32 12)(3 19 29 25)(4 23 30 10)(5 11 16 24)(6 28 13 18)(7 9 14 22)(8 26 15 20)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O2P2Q4A···4P4Q4R4S4T
order12···22···2224···44444
size11···12···2444···48888

38 irreducible representations

dim111111224
type++++++++
imageC1C2C2C2C2C2D4C4○D42+ 1+4
kernelC24.96D4C243C4C23.34D4C23.8Q8C23.11D4C22×C22⋊C4C24C23C22
# reps1144424162

Matrix representation of C24.96D4 in GL6(𝔽5)

100000
040000
001000
000400
000010
000004
,
400000
040000
004000
000400
000010
000004
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
030000
200000
000300
003000
000002
000030
,
200000
030000
003000
000300
000001
000010

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24.96D4 in GAP, Magma, Sage, TeX

C_2^4._{96}D_4
% in TeX

G:=Group("C2^4.96D4");
// GroupNames label

G:=SmallGroup(128,1215);
// by ID

G=gap.SmallGroup(128,1215);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,100,675]);
// Polycyclic

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

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