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G = C24.90D4  order 128 = 27

45th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.90D4, C25.17C22, C23.190C24, C24.539C23, C22.302+ (1+4), C24⋊9(C2×C4), C24⋊3C4⋊4C2, (C22×D4)⋊21C4, (C23×C4)⋊4C22, (D4×C23).7C2, C23⋊5(C22⋊C4), C23.363(C2×D4), C23.23D4⋊3C2, C23.34D4⋊9C2, C2.1(C23⋊3D4), C22.81(C23×C4), C23.78(C22×C4), C22.84(C22×D4), C2.C42⋊7C22, (C22×C4).455C23, C2.5(C22.11C24), (C22×D4).472C22, (C2×D4)⋊39(C2×C4), (C22×C4)⋊19(C2×C4), (C22×C22⋊C4)⋊6C2, (C2×C22⋊C4)⋊3C22, (C2×C4).214(C22×C4), C2.12(C22×C22⋊C4), C22.15(C2×C22⋊C4), SmallGroup(128,1040)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C24.90D4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C24.90D4
Lower central
C1 — C22 — C24.90D4
Upper central
C1 — C23 — C24.90D4
Jennings
C1 — C23 — C24.90D4

Subgroups: 1260 in 580 conjugacy classes, 180 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×16], C4 [×12], C22, C22 [×18], C22 [×88], C2×C4 [×4], C2×C4 [×52], D4 [×32], C23, C23 [×38], C23 [×80], C22⋊C4 [×24], C22×C4 [×14], C22×C4 [×20], C2×D4 [×16], C2×D4 [×48], C24, C24 [×20], C24 [×12], C2.C42 [×8], C2×C22⋊C4 [×16], C2×C22⋊C4 [×8], C23×C4, C23×C4 [×4], C22×D4 [×12], C22×D4 [×8], C25 [×2], C24⋊3C4 [×2], C23.34D4 [×2], C23.23D4 [×8], C22×C22⋊C4 [×2], D4×C23, C24.90D4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], 2+ (1+4) [×4], C22×C22⋊C4, C22.11C24 [×2], C23⋊3D4 [×4], C24.90D4

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00400000
00040000
00001200
00000400
00000043
00000001
,
40000000
04000000
00100000
00010000
00004000
00000400
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
03000000
20000000
00010000
00400000
00000010
00000044
00004000
00001100
,
02000000
20000000
00010000
00100000
00000010
00000001
00001000
00000100

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

44 conjugacy classes

class 1 2A···2G2H···2S2T2U2V2W4A···4T
order12···22···222224···4
size11···12···244444···4

44 irreducible representations

dim111111124
type++++++++
imageC1C2C2C2C2C2C4D42+ (1+4)
kernelC24.90D4C24⋊3C4C23.34D4C23.23D4C22×C22⋊C4D4×C23C22×D4C24C22
# reps1228211684

In GAP, Magma, Sage, TeX 

C_2^4._{90}D_4
 
% in TeX
 
G:=Group("C2^4.90D4");
 
// GroupNames label
 
G:=SmallGroup(128,1040);
 
// by ID
 
G=gap.SmallGroup(128,1040);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,675]);
 
// 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,a*e=e*a,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=d*e^-1>;
 
// generators/relations
 

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