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D₄×D₁₇ #63253

order	272
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

D₄×D₁₇

Splittings

Fitting	D₄×C₁₇ ⋊ C₂
Sylow	C₁₇ ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₇
commutator, socle, cosocle	C₂×C₁₇	C₂³
Fitting	D₄×C₁₇	C₂
hypercenter	D₄	D₁₇
hypocenter	C₁₇	C₂×D₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ D₄

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 68

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10587547, D₈×D₁₇, #10587597, …
maximal normal subgroup of	#10587547, D₈×D₁₇, #10587597, …
maximal quotient of	#10587564, C₂²⋊C₄×D₁₇, #10587567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(272, 40);