C₇×#577 #90598

order	448
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#577

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₇×#41 ◁ G

258 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11741457, #11741458, #11741459, …
maximal subgroup of	#11741457, #11741458, #11741459, …
maximal normal subgroup of	#11741457, #11741458, #11741459, …
maximal quotient of	C₇×#2897, C₇×#2898, C₇×#2912, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1360);