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G = C32.C4order 128 = 27

1st non-split extension by C32 of C4 acting via C4/C2=C2

p-group, metacyclic, nilpotent (class 5), monomial

Aliases: C32.1C4, C16.4Q8, C4.19D16, C8.10Q16, C22.1Q32, (C2×C32).5C2, (C2×C4).70D8, C8.17(C4⋊C4), C16.18(C2×C4), (C2×C8).266D4, C2.5(C163C4), C4.12(C2.D8), C8.4Q8.1C2, (C2×C16).93C22, SmallGroup(128,157)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — C32.C4
C1C2C4C8C2×C8C2×C16C2×C32 — C32.C4
C1C2C4C8C16 — C32.C4
C1C4C2×C4C2×C8C2×C16 — C32.C4
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C2×C16 — C32.C4

Generators and relations for C32.C4
 G = < a,b | a32=1, b4=a16, bab-1=a15 >

2C2
8C8
8C8
4M4(2)
4M4(2)
2C8.C4
2C8.C4

Smallest permutation representation of C32.C4
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 51 9 43 17 35 25 59)(2 34 10 58 18 50 26 42)(3 49 11 41 19 33 27 57)(4 64 12 56 20 48 28 40)(5 47 13 39 21 63 29 55)(6 62 14 54 22 46 30 38)(7 45 15 37 23 61 31 53)(8 60 16 52 24 44 32 36)

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

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

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

38 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F8G8H16A···16H32A···32P
order1224448888888816···1632···32
size1121122222161616162···22···2

38 irreducible representations

dim11112222222
type+++-+-++-
imageC1C2C2C4Q8D4Q16D8D16Q32C32.C4
kernelC32.C4C8.4Q8C2×C32C32C16C2×C8C8C2×C4C4C22C1
# reps121411224416

Matrix representation of C32.C4 in GL2(𝔽97) generated by

300
042
,
01
750
G:=sub<GL(2,GF(97))| [30,0,0,42],[0,75,1,0] >;

C32.C4 in GAP, Magma, Sage, TeX

C_{32}.C_4
% in TeX

G:=Group("C32.C4");
// GroupNames label

G:=SmallGroup(128,157);
// by ID

G=gap.SmallGroup(128,157);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,148,422,268,248,1684,242,4037,124]);
// Polycyclic

G:=Group<a,b|a^32=1,b^4=a^16,b*a*b^-1=a^15>;
// generators/relations

Export

Subgroup lattice of C32.C4 in TeX

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