#89291

order	448
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #362

Subgroups

	subgroup	G/H
center	C₂	#6383
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	#102
prime radical	C₂²×C₇	SD₁₆
cosocle	C₂×C₇×C₈	C₂²
hypercenter	#108	D₇
hypocenter	C₇	#362
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,4,6,7,5,3)(8,10,11,19,13,22,25,21)(9,14,15,30,17,32,23,31)(12,27,24,29,28,38,33,20)(16,34,26,37,35,36,39,18);(2,3)(4,5)(6,7)(8,9,13,17)(10,18,22,37)(11,23,25,15)(12,26,28,39)(14,29,32,20)(16,33,35,24)(19,36,21,34)(27,31,38,30)

Polycyclic presentation:

2^6,7|a^2=f,b^2=c,c^2=f,d^2=ef,e^2=f,b^a=bd,c^a=cf,d^a=de,e^a=ef,g^a=g^6,d^b=def,e^b=ef

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726997, #11726999, #11727010, …
maximal normal subgroup of	#11726997, #11726999, #11727010, …
maximal quotient of	#11725272, #11725280, #11725281, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 53);