Copied to
clipboard

G = C24.24D4order 128 = 27

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

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

Aliases: C24.24D4, C4⋊C432D4, C4.88(C4×D4), C41D411C4, C4211(C2×C4), (C2×D4).74D4, C4.7(C4⋊D4), C426C423C2, C2.5(D44D4), C23.562(C2×D4), C22.11C241C2, C22.101C22≀C2, C23.37D422C2, 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×C41D4).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

C1C2×C4 — C24.24D4
C1C2C4C2×C4C22×C4C22×D4C22.11C24 — C24.24D4
C1C2C2×C4 — C24.24D4
C1C22C22×C4 — C24.24D4
C1C2C2C22×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 [×2], C2 [×8], C4 [×4], C4 [×8], C22 [×3], C22 [×26], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×30], C23, C23 [×4], C23 [×16], C42 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×D4 [×25], C24 [×2], C24, C4.D4 [×2], D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×4], C41D4 [×4], C41D4 [×2], C2×M4(2) [×2], C22×D4 [×2], C22×D4 [×2], C426C4 [×2], C2×C4.D4, C23.37D4 [×2], C22.11C24, C2×C41D4, C24.24D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D44D4 [×2], 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○D4D44D4
kernelC24.24D4C426C4C2×C4.D4C23.37D4C22.11C24C2×C41D4C41D4C4⋊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

׿
×
𝔽