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C₂×D₇×C₁₆ #89671

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₇×C₁₆

Splittings

Fitting	C₂×C₇×C₁₆ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂²×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₇
commutator, hypocenter	C₇	C₂²×C₁₆
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂×C₈
prime radical	C₂²×D₇	C₈
cosocle	C₇×C₈	C₂³
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₆

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2)(4,5)(6,7)(8,9);(3,4,6,8,9,7,5)(10,11,12,15,13,16,18,21,14,17,19,22,20,23,24,25);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×D₇)⋊C₃₂, C₄×D₇×C₁₆, #11726805, …
maximal normal subgroup of	(C₂×D₇)⋊C₃₂, C₄×D₇×C₁₆, #11726805, …
maximal quotient of	C₁₆×Dic₁₄, C₄×D₇×C₁₆, #11726806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 433);