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M₁₆ #34

order	16
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×D₄ (order 16) · Inn(G) = C₂² · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #23872
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

modular maximal cyclic

M₁₆

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄	C₂²
commutator, socle	C₂	C₂×C₄
prime radical	C₂²	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(1,2,3,4,5,6,7,8);(2,6)(4,8)

Polycyclic presentation:

2^4|a^2=c,c^2=d,b^a=bd

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	1	1	1	2	2	2	2	2	2
order	1	2	4	4	2	4	8	8	8	8
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	-1	-1	-1	-1
χ₃	1	1	1	1	-1	-1	1	1	-1	-1
χ₄	1	1	1	1	-1	-1	-1	-1	1	1
χ₅	1	1	-1	-1	1	-1	$i$	$- i$	$i$	$- i$
χ₆	1	1	-1	-1	1	-1	$- i$	$i$	$- i$	$i$
χ₇	1	1	-1	-1	-1	1	$i$	$- i$	$- i$	$i$
χ₈	1	1	-1	-1	-1	1	$- i$	$i$	$i$	$- i$
χ₉	2	-2	$2 i$	$- 2 i$	0	0	0	0	0	0
χ₁₀	2	-2	$- 2 i$	$2 i$	0	0	0	0	0	0

Here i is the imaginary unit (i² = −1).

Relationships

Fitting subgroup of	M₁₆×A₅, #11751543, #14514736, …
maximal subgroup of	#100, #101, #104, …
maximal normal subgroup of	#100, #101, #104, …
maximal quotient of	#97, C₂²⋊C₈, C₄⋊C₈, …
outer automorphism group of	C₅²⋊C₃, C₂×C₅²⋊C₃, H(3,𝔽₅)⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16, 6);