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G = C24.91D4order 128 = 27

46th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.91D4, C24.12Q8, C25.20C22, C24.544C23, C23.197C24, C22.362+ 1+4, C234(C4⋊C4), C24.69(C2×C4), C23.91(C2×Q8), C23.604(C2×D4), C23.8Q83C2, C2.2(C233D4), C22.88(C23×C4), (C23×C4).42C22, C23.7Q813C2, C2.1(C232Q8), C22.88(C22×D4), C22.30(C22×Q8), C23.122(C22×C4), (C22×C4).462C23, C2.C429C22, C2.9(C22.11C24), (C2×C4⋊C4)⋊6C22, C22⋊C437(C2×C4), (C2×C22⋊C4)⋊22C4, (C22×C4)⋊21(C2×C4), C22.29(C2×C4⋊C4), C2.11(C22×C4⋊C4), (C2×C4).220(C22×C4), (C22×C22⋊C4).11C2, (C2×C22⋊C4).424C22, SmallGroup(128,1047)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.91D4
C1C2C22C23C24C23×C4C22×C22⋊C4 — C24.91D4
C1C22 — C24.91D4
C1C23 — C24.91D4
C1C23 — C24.91D4

Generators and relations for C24.91D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, eae-1=ac=ca, ad=da, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 828 in 408 conjugacy classes, 180 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×16], C22 [×3], C22 [×16], C22 [×52], C2×C4 [×8], C2×C4 [×56], C23, C23 [×34], C23 [×36], C22⋊C4 [×16], C22⋊C4 [×16], C4⋊C4 [×8], C22×C4 [×20], C22×C4 [×20], C24, C24 [×14], C24 [×4], C2.C42 [×8], C2×C22⋊C4 [×20], C2×C22⋊C4 [×8], C2×C4⋊C4 [×8], C23×C4 [×6], C25, C23.7Q8 [×4], C23.8Q8 [×8], C22×C22⋊C4, C22×C22⋊C4 [×2], C24.91D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, 2+ 1+4 [×4], C22×C4⋊C4, C22.11C24 [×2], C233D4 [×2], C232Q8 [×2], C24.91D4

Smallest permutation representation of C24.91D4
On 32 points
Generators in S32
(1 15)(2 18)(3 13)(4 20)(5 17)(6 16)(7 19)(8 14)(9 25)(10 23)(11 27)(12 21)(22 30)(24 32)(26 31)(28 29)
(1 13)(2 14)(3 15)(4 16)(5 19)(6 20)(7 17)(8 18)(9 24)(10 21)(11 22)(12 23)(25 32)(26 29)(27 30)(28 31)
(1 5)(2 6)(3 7)(4 8)(9 30)(10 31)(11 32)(12 29)(13 19)(14 20)(15 17)(16 18)(21 28)(22 25)(23 26)(24 27)
(1 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 25)(10 26)(11 27)(12 28)(21 29)(22 30)(23 31)(24 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 32 15 24)(2 31 16 23)(3 30 13 22)(4 29 14 21)(5 11 17 27)(6 10 18 26)(7 9 19 25)(8 12 20 28)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2S4A···4X
order12···22···24···4
size11···12···24···4

44 irreducible representations

dim11111224
type+++++-+
imageC1C2C2C2C4D4Q82+ 1+4
kernelC24.91D4C23.7Q8C23.8Q8C22×C22⋊C4C2×C22⋊C4C24C24C22
# reps148316444

Matrix representation of C24.91D4 in GL8(𝔽5)

40000000
04000000
00100000
00010000
00001000
00000400
00000040
00000001
,
10000000
01000000
00400000
00040000
00004000
00000400
00000410
00004001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
40000000
00040000
00100000
00000100
00004000
00000004
00000010
,
40000000
01000000
00300000
00020000
00004002
00000130
00000040
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,1] >;

C24.91D4 in GAP, Magma, Sage, TeX

C_2^4._{91}D_4
% in TeX

G:=Group("C2^4.91D4");
// GroupNames label

G:=SmallGroup(128,1047);
// by ID

G=gap.SmallGroup(128,1047);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,184,675]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*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^-1>;
// generators/relations

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