#89572

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#103 ⋊ C₂
Sylow	C₇ ⋊ #488

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#103	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₂₈	C₂²
hypercenter	#103	D₇
hypocenter	C₇	#488
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,4,6,7,5,3)(9,13)(12,14)(16,18,20,26)(17,22,21,24)(19,28,27,25)(23,30,29,31);(8,10,11,15)(9,12,13,14)(17,21)(18,25)(19,27)(22,30)(24,31)(26,28);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12)(16,17)(18,24)(19,23)(20,21)(22,26)(25,30)(27,29)(28,31)

Polycyclic presentation:

2^6,7|a^2=e,b^2=e,c^2=f,d^2=f,b^a=bef,c^a=cef,g^a=g^6,c^b=cd,d^b=df,d^c=df

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#89572, #11736230, #11736252, …
maximal normal subgroup of	C₂×#89572, #11736230, #11736252, …
maximal quotient of	#11726558, #11726598, #11726600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 334);