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C₄×D₄₅ #67096

order	360
isoclinism	isoclinic class of D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₄₅

Splittings

Fitting	C₄×C₅×C₉ ⋊ C₂
derived, Hall	C₅×C₉ ⋊ C₂×C₄
Sylow	C₉ ⋊ C₄×D₅
Sylow	C₅ ⋊ C₄×D₉

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₄₅
commutator, hypocenter	C₅×C₉	C₂×C₄
Frattini	C₂×C₃	C₂×D₁₅
Fitting	C₄×C₅×C₉	C₂
socle	C₂×C₃×C₅	C₂×D₃
prime radical	C₂×D₄₅	C₂
cosocle	C₂×C₅×C₉	C₂²
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10623539, #10623541, C₈×D₄₅, …
maximal normal subgroup of	#10623539, #10623541, C₈×D₄₅, …
maximal quotient of	C₈×D₄₅, #10623598, C₄×Dic₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 26);