#89665

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#108 ⋊ C₂
Sylow	C₇ ⋊ #430

Subgroups

	subgroup	G/H
center	C₄	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	C₂×C₄⋊C₄
prime radical	C₂²×D₇	Q₈
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#108	D₇
hypocenter	C₇	#430
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11727011, #11727021, #11727067, …
maximal normal subgroup of	#11727011, #11727021, #11727067, …
maximal quotient of	#11726282, #11726285, #11726286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 427);