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G = C162M5(2)  order 128 = 27

Central product of C16 and M5(2)

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

Aliases: C162M5(2), C162M4(2), C8.18C42, M5(2)⋊13C4, M4(2).5C8, C16(C4⋊C4), (C4×C16)⋊15C2, (C2×C16)⋊17C4, C4⋊C4.12C8, C8.12(C2×C8), C4.10(C4×C8), C16(C8⋊C4), C16(C22⋊C4), C16.26(C2×C4), C16(C165C4), C165C415C2, C22⋊C4.7C8, M4(2)(C2×C16), C16(C2×M5(2)), C16(C2×M4(2)), (C2×C16)M5(2), C8⋊C4.21C4, C2.1(D4○C16), C8.67(C22×C4), (C2×C4).74C42, C4.36(C22×C8), C22.10(C4×C8), C23.20(C2×C8), C4.37(C2×C42), C16(C42⋊C2), (C2×C8).622C23, (C4×C8).434C22, (C22×C16).17C2, C42.244(C2×C4), C16(C82M4(2)), C42⋊C2.38C4, (C2×C16).105C22, (C2×M4(2)).37C4, (C2×M5(2)).28C2, C22.25(C22×C8), C82M4(2).25C2, (C22×C8).576C22, C2.12(C2×C4×C8), C8⋊C4(C2×C16), (C2×C4).40(C2×C8), (C2×C8).246(C2×C4), (C2×C16)(C165C4), (C2×C16)(C42⋊C2), (C22×C4).407(C2×C4), (C2×C4).607(C22×C4), (C2×C16)(C82M4(2)), SmallGroup(128,840)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C162M5(2)
C1C2C4C2×C4C2×C8C22×C8C82M4(2) — C162M5(2)
C1C2 — C162M5(2)
C1C2×C16 — C162M5(2)
C1C2C2C2C2C4C4C2×C8 — C162M5(2)

Generators and relations for C162M5(2)
 G = < a,b,c | a16=c2=1, b4=a12, ab=ba, ac=ca, cbc=a8b >

Subgroups: 100 in 92 conjugacy classes, 84 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×8], C23, C16 [×8], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C2×C16 [×2], C2×C16 [×6], M5(2) [×4], C42⋊C2, C22×C8, C2×M4(2), C4×C16 [×2], C165C4 [×2], C82M4(2), C22×C16, C2×M5(2), C162M5(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], C2×C42, C22×C8 [×2], C2×C4×C8, D4○C16 [×2], C162M5(2)

Smallest permutation representation of C162M5(2)
On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 50 42 28 13 62 38 24 9 58 34 20 5 54 46 32)(2 51 43 29 14 63 39 25 10 59 35 21 6 55 47 17)(3 52 44 30 15 64 40 26 11 60 36 22 7 56 48 18)(4 53 45 31 16 49 41 27 12 61 37 23 8 57 33 19)
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)

G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,50,42,28,13,62,38,24,9,58,34,20,5,54,46,32)(2,51,43,29,14,63,39,25,10,59,35,21,6,55,47,17)(3,52,44,30,15,64,40,26,11,60,36,22,7,56,48,18)(4,53,45,31,16,49,41,27,12,61,37,23,8,57,33,19), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)>;

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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,50,42,28,13,62,38,24,9,58,34,20,5,54,46,32)(2,51,43,29,14,63,39,25,10,59,35,21,6,55,47,17)(3,52,44,30,15,64,40,26,11,60,36,22,7,56,48,18)(4,53,45,31,16,49,41,27,12,61,37,23,8,57,33,19), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64) );

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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,50,42,28,13,62,38,24,9,58,34,20,5,54,46,32),(2,51,43,29,14,63,39,25,10,59,35,21,6,55,47,17),(3,52,44,30,15,64,40,26,11,60,36,22,7,56,48,18),(4,53,45,31,16,49,41,27,12,61,37,23,8,57,33,19)], [(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)])

80 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8H8I···8T16A···16P16Q···16AN
order12222244444···48···88···816···1616···16
size11112211112···21···12···21···12···2

80 irreducible representations

dim111111111111112
type++++++
imageC1C2C2C2C2C2C4C4C4C4C4C8C8C8D4○C16
kernelC162M5(2)C4×C16C165C4C82M4(2)C22×C16C2×M5(2)C8⋊C4C2×C16M5(2)C42⋊C2C2×M4(2)C22⋊C4C4⋊C4M4(2)C2
# reps12211148822881616

Matrix representation of C162M5(2) in GL3(𝔽17) generated by

100
060
006
,
1300
0014
030
,
1600
010
0016
G:=sub<GL(3,GF(17))| [1,0,0,0,6,0,0,0,6],[13,0,0,0,0,3,0,14,0],[16,0,0,0,1,0,0,0,16] >;

C162M5(2) in GAP, Magma, Sage, TeX

C_{16}\circ_2M_5(2)
% in TeX

G:=Group("C16o2M5(2)");
// GroupNames label

G:=SmallGroup(128,840);
// by ID

G=gap.SmallGroup(128,840);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,120,723,136,124]);
// Polycyclic

G:=Group<a,b,c|a^16=c^2=1,b^4=a^12,a*b=b*a,a*c=c*a,c*b*c=a^8*b>;
// generators/relations

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