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G = C8.5C42  order 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.Q8⋊5C4, C8⋊C4⋊6C4, C4.2(C4×Q8), C2.D8⋊10C4, C8.36(C4⋊C4), (C2×C8).29Q8, C8.C4⋊5C4, (C2×C8).386D4, C4.172(C4×D4), C22.2(C4×Q8), C4.22(C2×C42), C22.94(C4×D4), C2.5(C8.26D4), C42⋊6C4.4C2, C42.133(C2×C4), M4(2).20(C2×C4), C23.202(C4○D4), C8○2M4(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

Derived series
C1 — C4 — C8.5C42
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×C8 — C8○2M4(2) — C8.5C42
Lower central
C1 — C2 — C4 — C8.5C42
Upper central
C1 — C2×C4 — C22×C8 — C8.5C42
Jennings
C1 — C2 — C2 — C22×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, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C8⋊C4, C4.Q8, C2.D8, C8.C4, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C42⋊6C4, C2×C8⋊C4, C8○2M4(2), C23.25D4, C2×C8.C4, C8.5C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×C4⋊C4, C8.26D4, 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 30)(2 14 19 29)(3 13 20 28)(4 12 21 27)(5 11 22 26)(6 10 23 25)(7 9 24 32)(8 16 17 31)

(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), (1,15,18,30)(2,14,19,29)(3,13,20,28)(4,12,21,27)(5,11,22,26)(6,10,23,25)(7,9,24,32)(8,16,17,31), (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), (1,15,18,30)(2,14,19,29)(3,13,20,28)(4,12,21,27)(5,11,22,26)(6,10,23,25)(7,9,24,32)(8,16,17,31), (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)], [(1,15,18,30),(2,14,19,29),(3,13,20,28),(4,12,21,27),(5,11,22,26),(6,10,23,25),(7,9,24,32),(8,16,17,31)], [(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
kernelC8.5C42C42⋊6C4C2×C8⋊C4C8○2M4(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
 

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