Copied to
clipboard

G = C8.5C42order 128 = 27

5th non-split extension by C8 of C42 acting via C42/C22=C22

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

Aliases: C8.5C42, C4.Q85C4, C8⋊C46C4, C4.2(C4×Q8), C2.D810C4, C8.36(C4⋊C4), (C2×C8).29Q8, C8.C45C4, (C2×C8).386D4, C4.172(C4×D4), C22.2(C4×Q8), C4.22(C2×C42), C22.94(C4×D4), C2.5(C8.26D4), C426C4.4C2, C42.133(C2×C4), M4(2).20(C2×C4), C23.202(C4○D4), C82M4(2).16C2, (C22×C8).209C22, (C2×C42).240C22, (C22×C4).1314C23, C23.25D4.11C2, C42⋊C2.266C22, C22.24(C42⋊C2), (C2×M4(2)).311C22, C2.12(C4×C4⋊C4), C4.77(C2×C4⋊C4), (C2×C8⋊C4).1C2, C4⋊C4.145(C2×C4), (C2×C8).139(C2×C4), (C2×C4).183(C2×Q8), (C2×C8.C4).8C2, (C2×C4).1508(C2×D4), (C2×C4).545(C4○D4), (C2×C4).523(C22×C4), SmallGroup(128,505)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.5C42
C1C2C22C23C22×C4C22×C8C82M4(2) — C8.5C42
C1C2C4 — C8.5C42
C1C2×C4C22×C8 — C8.5C42
C1C2C2C22×C4 — C8.5C42

Generators and relations for C8.5C42
 G = < a,b,c | a8=b4=c4=1, bab-1=a-1, cac-1=a5, cbc-1=a6b >

Subgroups: 164 in 112 conjugacy classes, 76 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C8 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×8], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×10], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4, C4×C8 [×2], C8⋊C4 [×4], C8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C8.C4 [×4], C2×C42, C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×2], C426C4 [×2], C2×C8⋊C4, C82M4(2) [×2], C23.25D4, C2×C8.C4, C8.5C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×C4⋊C4, C8.26D4 [×2], C8.5C42

Smallest permutation representation of C8.5C42
On 32 points
Generators in S32
(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 15 18 26)(2 14 19 25)(3 13 20 32)(4 12 21 31)(5 11 22 30)(6 10 23 29)(7 9 24 28)(8 16 17 27)
(2 6)(4 8)(9 15 13 11)(10 12 14 16)(17 21)(19 23)(25 27 29 31)(26 32 30 28)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4R8A···8H8I···8T
order1222224444444···48···88···8
size1111221111224···42···24···4

44 irreducible representations

dim111111111122224
type+++++++-
imageC1C2C2C2C2C2C4C4C4C4D4Q8C4○D4C4○D4C8.26D4
kernelC8.5C42C426C4C2×C8⋊C4C82M4(2)C23.25D4C2×C8.C4C8⋊C4C4.Q8C2.D8C8.C4C2×C8C2×C8C2×C4C23C2
# reps121211844822224

Matrix representation of C8.5C42 in GL6(𝔽17)

400000
2130000
000200
002000
00120129
0013105
,
13160000
040000
000010
007071
001000
00101100
,
400000
2130000
001000
0001600
000040
001601213

G:=sub<GL(6,GF(17))| [4,2,0,0,0,0,0,13,0,0,0,0,0,0,0,2,12,1,0,0,2,0,0,3,0,0,0,0,12,10,0,0,0,0,9,5],[13,0,0,0,0,0,16,4,0,0,0,0,0,0,0,7,1,10,0,0,0,0,0,1,0,0,1,7,0,10,0,0,0,1,0,0],[4,2,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,16,0,0,0,16,0,0,0,0,0,0,4,12,0,0,0,0,0,13] >;

C8.5C42 in GAP, Magma, Sage, TeX

C_8._5C_4^2
% in TeX

G:=Group("C8.5C4^2");
// GroupNames label

G:=SmallGroup(128,505);
// by ID

G=gap.SmallGroup(128,505);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,1018,248,1411]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^6*b>;
// generators/relations

׿
×
𝔽