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G = C42⋊1C8  order 128 = 27

1st semidirect product of C42 and C8 acting via C8/C2=C4

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42⋊1C8, C42.1Q8, C42.19D4, (C4×C8)⋊1C4, C4.1(C4⋊C8), C4.16(C4×C8), (C2×C42).4C4, (C2×C4).78C42, C4.11(C8⋊C4), C42⋊4C4.3C2, C42.288(C2×C4), (C22×C4).116D4, (C2×C4).63M4(2), C2.1(C4.9C42), C22.8(C22⋊C8), C42.12C4.1C2, (C2×C42).121C22, C23.133(C22⋊C4), C2.3(C22.7C42), C22.15(C2.C42), (C2×C4).66(C2×C8), (C2×C4).92(C4⋊C4), (C22×C4).460(C2×C4), (C2×C4).287(C22⋊C4), SmallGroup(128,6)

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

Derived series
C1 — C4 — C42⋊1C8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊4C4 — C42⋊1C8
Lower central
C1 — C4 — C42⋊1C8
Upper central
C1 — C2×C4 — C42⋊1C8
Jennings
C1 — C22 — C22 — C2×C42 — C42⋊1C8

Generators and relations for C42⋊1C8
 G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab-1, bc=cb >

Subgroups: 152 in 84 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C42, C2×C8, C22×C4, C22×C4, C22×C4, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊4C4, C42.12C4, C42⋊1C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C22.7C42, C4.9C42, C42⋊1C8

Smallest permutation representation of C42⋊1C8
►On 32 points
Generators in S32
(1 5)(2 16 32 18)(3 29)(4 20 26 10)(6 12 28 22)(7 25)(8 24 30 14)(9 23)(11 15)(13 19)(17 21)(27 31)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N4O···4V8A···8P
order12222244444···44···48···8
size11112211112···24···44···4

44 irreducible representations

dim11111122224
type++++-+
imageC1C2C2C4C4C8D4Q8D4M4(2)C4.9C42
kernelC42⋊1C8C42⋊4C4C42.12C4C4×C8C2×C42C42C42C42C22×C4C2×C4C2
# reps112841611244

Matrix representation of C42⋊1C8 ►in GL6(𝔽17)

1600000
1610000
0016000
0016100
0000130
0000134
,
1600000
0160000
004000
000400
000040
000004
,
810000
090000
000010
000001
0011500
0011600

G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[8,0,0,0,0,0,1,9,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C42⋊1C8 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_1C_8
 
% in TeX
 
G:=Group("C4^2:1C8");
 
// GroupNames label
 
G:=SmallGroup(128,6);
 
// by ID
 
G=gap.SmallGroup(128,6);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,184,570,248,1684]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b>;
 
// generators/relations
 

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