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G = C24.91D4  order 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, C23.197C24, C24.544C23, C22.362+ (1+4), C23⋊4(C4⋊C4), C24.69(C2×C4), C23.91(C2×Q8), C23.604(C2×D4), C23.8Q8⋊3C2, C2.2(C23⋊3D4), (C23×C4).42C22, C22.88(C23×C4), C23.7Q8⋊13C2, C2.1(C23⋊2Q8), C22.88(C22×D4), C22.30(C22×Q8), C2.C42⋊9C22, C23.122(C22×C4), (C22×C4).462C23, C2.9(C22.11C24), (C2×C4⋊C4)⋊6C22, C22⋊C4⋊37(C2×C4), (C2×C22⋊C4)⋊22C4, (C22×C4)⋊21(C2×C4), C2.11(C22×C4⋊C4), C22.29(C2×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

Derived series
C1 — C22 — C24.91D4
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C24.91D4
Lower central
C1 — C22 — C24.91D4
Upper central
C1 — C23 — C24.91D4
Jennings
C1 — C23 — C24.91D4

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], C23⋊3D4 [×2], C23⋊2Q8 [×2], C24.91D4

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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