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G = C8⋊C4⋊17C4  order 128 = 27

12nd semidirect product of C8⋊C4 and C4 acting via C4/C2=C2

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

Aliases: (C4×C8)⋊9C4, C8⋊C4⋊17C4, C4.9C42.4C2, C42⋊6C4.5C2, C42.322(C2×C4), C2.5(C42⋊5C4), (C4×M4(2)).23C2, C4.62(C42⋊C2), C4.12(C42⋊2C2), C4.10C42.3C2, C23.116(C4○D4), M4(2)⋊4C4.4C2, (C2×C42).257C22, (C22×C4).671C23, C42⋊C2.7C22, C22.4(C42⋊2C2), C22.33(C42⋊C2), (C2×M4(2)).315C22, (C2×C8).14(C2×C4), (C2×C4).314(C4○D4), (C2×C4).541(C22×C4), SmallGroup(128,573)

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

Derived series
C1 — C2×C4 — C8⋊C4⋊17C4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — C4×M4(2) — C8⋊C4⋊17C4
Lower central
C1 — C2 — C2×C4 — C8⋊C4⋊17C4
Upper central
C1 — C4 — C2×M4(2) — C8⋊C4⋊17C4
Jennings
C1 — C2 — C2 — C22×C4 — C8⋊C4⋊17C4

Generators and relations for C8⋊C4⋊17C4
 G = < a,b,c | a8=b4=c4=1, bab-1=a5, cac-1=ab2, cbc-1=a2b-1 >

Subgroups: 148 in 81 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C2×C42, C42⋊C2, C2×M4(2), C2×M4(2), C4.9C42, C4.10C42, C42⋊6C4, M4(2)⋊4C4, C4×M4(2), C8⋊C4⋊17C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C42⋊C2, C42⋊2C2, C42⋊5C4, C8⋊C4⋊17C4

Permutation representations of C8⋊C4⋊17C4
►On 16 points - transitive group 16T294
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 6)(4 8)(9 11 13 15)(10 16 14 12)

(1 15)(2 16 6 12)(3 9)(4 10 8 14)(5 11)(7 13)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (1,15)(2,16,6,12)(3,9)(4,10,8,14)(5,11)(7,13)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (1,15)(2,16,6,12)(3,9)(4,10,8,14)(5,11)(7,13) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,11,13,15),(10,16,14,12)], [(1,15),(2,16,6,12),(3,9),(4,10,8,14),(5,11),(7,13)]])
 
G:=TransitiveGroup(16,294);
 

32 conjugacy classes

class 1 2A2B2C2D4A4B4C···4I4J4K4L4M4N4O8A···8H8I8J8K8L
order12222444···44444448···88888
size11222112···24488884···48888

32 irreducible representations

dim11111111224
type++++++
imageC1C2C2C2C2C2C4C4C4○D4C4○D4C8⋊C4⋊17C4
kernelC8⋊C4⋊17C4C4.9C42C4.10C42C42⋊6C4M4(2)⋊4C4C4×M4(2)C4×C8C8⋊C4C2×C4C23C1
# reps111221441024

Matrix representation of C8⋊C4⋊17C4 ►in GL4(𝔽5) generated by

0200
1000
4004
0330
,
4001
0210
0300
4004
,
0002
0400
0210
2000
G:=sub<GL(4,GF(5))| [0,1,4,0,2,0,0,3,0,0,0,3,0,0,4,0],[4,0,0,4,0,2,3,0,0,1,0,0,1,0,0,4],[0,0,0,2,0,4,2,0,0,0,1,0,2,0,0,0] >;
 

C8⋊C4⋊17C4 in GAP, Magma, Sage, TeX 

C_8\rtimes C_4\rtimes_{17}C_4
 
% in TeX
 
G:=Group("C8:C4:17C4");
 
// GroupNames label
 
G:=SmallGroup(128,573);
 
// by ID
 
G=gap.SmallGroup(128,573);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,176,422,58,2019,248,172,1027]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^5,c*a*c^-1=a*b^2,c*b*c^-1=a^2*b^-1>;
 
// generators/relations
 

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