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G = C421C8order 128 = 27

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

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

Aliases: C421C8, 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), C424C4.3C2, 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

C1C4 — C421C8
C1C2C22C23C22×C4C2×C42C424C4 — C421C8
C1C4 — C421C8
C1C2×C4 — C421C8
C1C22C22C2×C42 — C421C8

Generators and relations for C421C8
 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 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], C23, C42 [×2], C42 [×6], C42 [×2], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2.C42 [×2], C4×C8 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C424C4, C42.12C4 [×2], C421C8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42, C4.9C42 [×2], C421C8

Smallest permutation representation of C421C8
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
kernelC421C8C424C4C42.12C4C4×C8C2×C42C42C42C42C22×C4C2×C4C2
# reps112841611244

Matrix representation of C421C8 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] >;

C421C8 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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