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G = C24.24D4  order 128 = 27

24th non-split extension by C24 of D4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24.24D4, C4⋊C4⋊32D4, C4.88(C4×D4), C4⋊1D4⋊11C4, C42⋊11(C2×C4), (C2×D4).74D4, C4.7(C4⋊D4), C42⋊6C4⋊23C2, C2.5(D4⋊4D4), C23.562(C2×D4), C22.11C24⋊1C2, C22.101C22≀C2, C23.37D4⋊22C2, C23.10(C22⋊C4), (C2×C42).286C22, (C22×C4).685C23, (C22×D4).20C22, C42⋊C2.22C22, C2.26(C23.23D4), (C2×M4(2)).182C22, C22.50(C22.D4), (C2×D4)⋊8(C2×C4), (C2×C4⋊1D4).4C2, (C2×C4).57(C4○D4), (C2×C4.D4)⋊16C2, (C2×C4).1006(C2×D4), (C2×C4).187(C22×C4), C22.42(C2×C22⋊C4), SmallGroup(128,619)

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

Derived series
C1 — C2×C4 — C24.24D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.11C24 — C24.24D4
Lower central
C1 — C2 — C2×C4 — C24.24D4
Upper central
C1 — C22 — C22×C4 — C24.24D4
Jennings
C1 — C2 — C2 — C22×C4 — C24.24D4

Generators and relations for C24.24D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, eae-1=faf=acd, bc=cb, bd=db, be=eb, bf=fb, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=be3 >

Subgroups: 580 in 222 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C4.D4, D4⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4⋊1D4, C4⋊1D4, C2×M4(2), C22×D4, C22×D4, C42⋊6C4, C2×C4.D4, C23.37D4, C22.11C24, C2×C4⋊1D4, C24.24D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, D4⋊4D4, C24.24D4

Permutation representations of C24.24D4
►On 16 points - transitive group 16T302
Generators in S16
(1 13)(2 10)(3 11)(4 16)(5 9)(6 14)(7 15)(8 12)

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

(2 6)(4 8)(10 14)(12 16)

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

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

(1 3)(2 10)(4 16)(5 7)(6 14)(8 12)(9 11)(13 15)
 
G:=sub<Sym(16)| (1,13)(2,10)(3,11)(4,16)(5,9)(6,14)(7,15)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (2,6)(4,8)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,3)(2,10)(4,16)(5,7)(6,14)(8,12)(9,11)(13,15)>;
 
G:=Group( (1,13)(2,10)(3,11)(4,16)(5,9)(6,14)(7,15)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (2,6)(4,8)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,3)(2,10)(4,16)(5,7)(6,14)(8,12)(9,11)(13,15) );
 
G=PermutationGroup([[(1,13),(2,10),(3,11),(4,16),(5,9),(6,14),(7,15),(8,12)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(2,6),(4,8),(10,14),(12,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,3),(2,10),(4,16),(5,7),(6,14),(8,12),(9,11),(13,15)]])
 
G:=TransitiveGroup(16,302);
 

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E···4P8A8B8C8D
order12222222222244444···48888
size11112244448822224···48888

32 irreducible representations

dim111111122224
type++++++++++
imageC1C2C2C2C2C2C4D4D4D4C4○D4D4⋊4D4
kernelC24.24D4C42⋊6C4C2×C4.D4C23.37D4C22.11C24C2×C4⋊1D4C4⋊1D4C4⋊C4C2×D4C24C2×C4C2
# reps121211842244

Matrix representation of C24.24D4 ►in GL6(𝔽17)

0160000
1600000
0016000
0016100
0000160
001001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
00160160
00160016
,
100000
010000
0016000
0001600
0000160
0000016
,
1300000
040000
00160150
001601616
000110
000010
,
1600000
010000
0011500
0001600
000110
00161016

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,0,0,0,0,0,1,0,0,0,15,16,1,1,0,0,0,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,15,16,1,1,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

C24.24D4 in GAP, Magma, Sage, TeX 

C_2^4._{24}D_4
 
% in TeX
 
G:=Group("C2^4.24D4");
 
// GroupNames label
 
G:=SmallGroup(128,619);
 
// by ID
 
G=gap.SmallGroup(128,619);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,1018,521,248,2804,1411,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=f*a*f=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=b*e^3>;
 
// generators/relations
 

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