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G = C8.16C42order 128 = 27

10th non-split extension by C8 of C42 acting via C42/C2×C4=C2

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

Aliases: C8.16C42, (C4×C8)⋊8C4, C8⋊C414C4, C8(C4.9C42), C4.42(C2×C42), C42.20(C2×C4), C4.9C42.6C2, C23.1(C4○D4), C8(C4.10C42), C2.6(C424C4), C4.45(C42⋊C2), C4.10C42.5C2, C82M4(2).14C2, (C22×C4).649C23, (C22×C8).376C22, C42⋊C2.259C22, C22.19(C42⋊C2), (C2×M4(2)).302C22, (C2×C8).9(C2×C4), (C2×C4).41(C4○D4), (C2×C4).520(C22×C4), SmallGroup(128,479)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.16C42
C1C2C22C23C22×C4C22×C8C82M4(2) — C8.16C42
C1C4 — C8.16C42
C1C8 — C8.16C42
C1C2C2C22×C4 — C8.16C42

Generators and relations for C8.16C42
 G = < a,b,c | a8=b4=c4=1, bab-1=cac-1=a5, cbc-1=a2b >

Subgroups: 148 in 100 conjugacy classes, 66 normal (9 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C4 [×6], C22 [×3], C22, C8, C8 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×6], C23, C42 [×6], C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×12], M4(2) [×6], C22×C4, C4×C8 [×6], C8⋊C4 [×6], C42⋊C2 [×3], C22×C8, C2×M4(2) [×3], C4.9C42 [×3], C4.10C42, C82M4(2) [×3], C8.16C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], C4○D4 [×4], C2×C42, C42⋊C2 [×6], C424C4, C8.16C42

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F···4Q8A8B8C8D8E···8J8K···8V
order12222444444···488888···88···8
size11222112224···411112···24···4

44 irreducible representations

dim111111224
type++++
imageC1C2C2C2C4C4C4○D4C4○D4C8.16C42
kernelC8.16C42C4.9C42C4.10C42C82M4(2)C4×C8C8⋊C4C2×C4C23C1
# reps13131212624

Matrix representation of C8.16C42 in GL4(𝔽17) generated by

2000
0200
00150
00015
,
0024
001515
8000
9900
,
0010
0001
161500
0100
G:=sub<GL(4,GF(17))| [2,0,0,0,0,2,0,0,0,0,15,0,0,0,0,15],[0,0,8,9,0,0,0,9,2,15,0,0,4,15,0,0],[0,0,16,0,0,0,15,1,1,0,0,0,0,1,0,0] >;

C8.16C42 in GAP, Magma, Sage, TeX

C_8._{16}C_4^2
% in TeX

G:=Group("C8.16C4^2");
// GroupNames label

G:=SmallGroup(128,479);
// by ID

G=gap.SmallGroup(128,479);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,58,248,1411,4037]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^2*b>;
// generators/relations

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