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G = C42⋊C23  order 128 = 27

4th semidirect product of C42 and C23 acting faithfully

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

Aliases: C24⋊4C23, C25⋊8C22, C42⋊4C23, C23.62C24, C22.121C25, C22.102+ 1+4, D42⋊18C2, C4⋊C4⋊10C23, D4⋊5D4⋊27C2, (C2×D4)⋊10C23, (C4×D4)⋊56C22, C23⋊3D4⋊9C2, (C22×C4)⋊5C23, (C2×Q8)⋊10C23, C4⋊1D4⋊20C22, C4⋊D4⋊32C22, C22⋊C4⋊12C23, (C2×C4).111C24, C22⋊Q8⋊40C22, C22≀C2⋊39C22, C24⋊C22⋊4C2, C42⋊2C2⋊8C22, C4.4D4⋊32C22, (C22×D4)⋊41C22, C42⋊C2⋊49C22, C22.54C24⋊2C2, C22.29C24⋊26C2, C22.32C24⋊10C2, C22.45C24⋊12C2, C2.50(C2×2+ 1+4), C22.D4⋊12C22, (C2×C22≀C2)⋊29C2, (C2×C4○D4)⋊40C22, (C2×C22⋊C4)⋊56C22, SmallGroup(128,2264)

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

Derived series
C1 — C22 — C42⋊C23
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C42⋊C23
Lower central
C1 — C22 — C42⋊C23
Upper central
C1 — C22 — C42⋊C23
Jennings
C1 — C22 — C42⋊C23

Generators and relations for C42⋊C23
 G = < a,b,c,d,e | a4=b4=c2=d2=e2=1, ab=ba, cac=dad=ab2, eae=a-1b2, cbc=b-1, dbd=a2b-1, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 1324 in 668 conjugacy classes, 384 normal (11 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, C24, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊1D4, C4⋊1D4, C22×D4, C2×C4○D4, C25, C2×C22≀C2, C23⋊3D4, C22.29C24, C22.32C24, D42, D4⋊5D4, C22.45C24, C22.54C24, C24⋊C22, C42⋊C23
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C42⋊C23

Permutation representations of C42⋊C23
►On 16 points - transitive group 16T206
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 9 13 7)(2 10 14 8)(3 11 15 5)(4 12 16 6)

(1 3)(2 16)(4 14)(5 9)(6 8)(7 11)(10 12)(13 15)

(1 13)(3 15)(5 7)(6 10)(8 12)(9 11)

(1 13)(2 4)(3 15)(5 11)(6 8)(7 9)(10 12)(14 16)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,13,7)(2,10,14,8)(3,11,15,5)(4,12,16,6), (1,3)(2,16)(4,14)(5,9)(6,8)(7,11)(10,12)(13,15), (1,13)(3,15)(5,7)(6,10)(8,12)(9,11), (1,13)(2,4)(3,15)(5,11)(6,8)(7,9)(10,12)(14,16)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,13,7)(2,10,14,8)(3,11,15,5)(4,12,16,6), (1,3)(2,16)(4,14)(5,9)(6,8)(7,11)(10,12)(13,15), (1,13)(3,15)(5,7)(6,10)(8,12)(9,11), (1,13)(2,4)(3,15)(5,11)(6,8)(7,9)(10,12)(14,16) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9,13,7),(2,10,14,8),(3,11,15,5),(4,12,16,6)], [(1,3),(2,16),(4,14),(5,9),(6,8),(7,11),(10,12),(13,15)], [(1,13),(3,15),(5,7),(6,10),(8,12),(9,11)], [(1,13),(2,4),(3,15),(5,11),(6,8),(7,9),(10,12),(14,16)]])
 
G:=TransitiveGroup(16,206);
 

38 conjugacy classes

class 1 2A2B2C2D···2I2J···2S4A···4R
order12222···22···24···4
size11112···24···44···4

38 irreducible representations

dim11111111114
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C22+ 1+4
kernelC42⋊C23C2×C22≀C2C23⋊3D4C22.29C24C22.32C24D42D4⋊5D4C22.45C24C22.54C24C24⋊C22C22
# reps13336363316

Matrix representation of C42⋊C23 ►in GL8(ℤ)

01-200000
-10020000
-10010000
01-100000
00000010
0000000-1
0000-1000
00000100
,
01000000
-10000000
000-10000
00100000
00000100
0000-1000
0000000-1
00000010
,
-10000000
01000000
00100000
000-10000
0000-1000
00000100
00000010
0000000-1
,
-10000000
01000000
00100000
000-10000
0000-1000
00000-100
00000010
00000001
,
-10000000
0-1000000
0-1100000
-10010000
0000-1000
00000-100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [0,-1,-1,0,0,0,0,0,1,0,0,1,0,0,0,0,-2,0,0,-1,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,-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],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-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,-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],[-1,0,0,-1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-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] >;
 

C42⋊C23 in GAP, Magma, Sage, TeX 

C_4^2\rtimes C_2^3
 
% in TeX
 
G:=Group("C4^2:C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2264);
 
// by ID
 
G=gap.SmallGroup(128,2264);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,2019,570,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^2=1,a*b=b*a,c*a*c=d*a*d=a*b^2,e*a*e=a^-1*b^2,c*b*c=b^-1,d*b*d=a^2*b^-1,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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