#89568

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#103	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#103	D₇
hypocenter	C₇	#479
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 448

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

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

Polycyclic presentation:

2^6,7|a^2=d,b^2=f,c^2=d,e^2=f,c^a=cdf,g^a=g^6,c^b=ce,e^b=ef,e^c=ef

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#89568, #11736224, #11736253, …
maximal normal subgroup of	C₂×#89568, #11736224, #11736253, …
maximal quotient of	#11726552, #11726555, #11726568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 330);