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C₂×C₄×C₁₇ #3630

order	136
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = D₄×C₁₆ (order 128) · Inn(G) = C₁ · Out(G) = D₄×C₁₆
holomorph	G ⋊ Aut(G) = B(4,𝔽₂)×AGL(1,𝔽₁₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

C₂×C₄×C₁₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₁₇
socle, prime radical	C₂²×C₁₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 136

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);(5,6)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

center of	C₁₇×#97, C₁₇×C₂²⋊C₈, C₁₇×C₄⋊C₈, …
derived subgroup of	#11762542, #11762543, #11762544, …
Fitting subgroup of	C₂×#3622, #63223, C₄×Dic₁₇, …
maximal subgroup of	C₂×#3622, #63223, C₄×Dic₁₇, …
maximal normal subgroup of	C₂×#3622, #63223, C₄×Dic₁₇, …
maximal quotient of	C₄²×C₁₇, C₂²⋊C₄×C₁₇, C₄⋊C₄×C₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(136, 9);