Copied to
clipboard

G = C42.242D4order 128 = 27

224th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.242D4, C42.368C23, (C4×C8)⋊65C22, C4⋊Q864C22, C4⋊C4.93C23, C8⋊C466C22, (C4×M4(2))⋊41C2, (C2×C4).338C24, (C2×C8).459C23, C23.680(C2×D4), (C22×C4).462D4, (C2×Q8).93C23, Q8⋊C455C22, C4.79(C4.4D4), (C2×D4).105C23, C42.C236C22, C23.38D437C2, (C2×C42).849C22, C22.598(C22×D4), D4⋊C4.133C22, C2.36(D8⋊C22), (C22×C4).1036C23, C4.4D4.136C22, C23.37D4.11C2, C22.33(C4.4D4), (C22×D4).370C22, (C22×Q8).303C22, C23.37C2311C2, C42.28C2231C2, C42⋊C2.143C22, C42.78C2227C2, (C2×M4(2)).375C22, C4.47(C2×C4○D4), (C2×C4).516(C2×D4), C2.49(C2×C4.4D4), (C2×C4).302(C4○D4), (C2×C4.4D4).40C2, SmallGroup(128,1872)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.242D4
C1C2C4C2×C4C22×C4C2×M4(2)C4×M4(2) — C42.242D4
C1C2C2×C4 — C42.242D4
C1C22C2×C42 — C42.242D4
C1C2C2C2×C4 — C42.242D4

Generators and relations for C42.242D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=a2c3 >

Subgroups: 404 in 200 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], D4 [×6], Q8 [×10], C23, C23 [×8], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×7], C24, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×4], C4.4D4 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×D4, C22×Q8, C4×M4(2), C23.37D4 [×2], C23.38D4 [×2], C42.78C22 [×4], C42.28C22 [×4], C2×C4.4D4, C23.37C23, C42.242D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D8⋊C22 [×2], C42.242D4

Smallest permutation representation of C42.242D4
On 32 points
Generators in S32
(1 21 29 15)(2 22 30 16)(3 23 31 9)(4 24 32 10)(5 17 25 11)(6 18 26 12)(7 19 27 13)(8 20 28 14)
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 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 32 25 8)(2 7 26 31)(3 30 27 6)(4 5 28 29)(9 12 19 22)(10 21 20 11)(13 16 23 18)(14 17 24 15)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K···4P8A···8H
order122222224···4444···48···8
size111122882···2448···84···4

32 irreducible representations

dim111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4D8⋊C22
kernelC42.242D4C4×M4(2)C23.37D4C23.38D4C42.78C22C42.28C22C2×C4.4D4C23.37C23C42C22×C4C2×C4C2
# reps112244112284

Matrix representation of C42.242D4 in GL6(𝔽17)

1680000
410000
0013000
0001300
0000130
0000013
,
1600000
0160000
0016200
0016100
0041301
00013160
,
1300000
0130000
0016080
0000413
0041310
000010
,
1300000
1640000
001090
00101313
0000160
00134160

G:=sub<GL(6,GF(17))| [16,4,0,0,0,0,8,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,4,0,0,0,2,1,13,13,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,4,0,0,0,0,0,13,0,0,0,8,4,1,1,0,0,0,13,0,0],[13,16,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,13,0,0,0,0,0,4,0,0,9,13,16,16,0,0,0,13,0,0] >;

C42.242D4 in GAP, Magma, Sage, TeX

C_4^2._{242}D_4
% in TeX

G:=Group("C4^2.242D4");
// GroupNames label

G:=SmallGroup(128,1872);
// by ID

G=gap.SmallGroup(128,1872);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,1018,521,2804,172,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations

׿
×
𝔽