#89301

order	448
isoclinism	isoclinic class of D₁₁₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #368

Subgroups

	subgroup	G/H
center	C₄	D₅₆
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle	C₂×C₇	#107
prime radical	C₂²×C₇	Q₁₆
cosocle	C₂×C₇×C₈	C₂²
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	#368
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

40 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)(8,10,20,36,13,23,27,37,14,24,25,29,11,21,12,22)(9,15,26,33,16,30,32,35,18,31,39,38,28,34,17,19);(2,3)(4,5)(6,7)(8,9,11,16,14,18,13,28)(10,19,21,33,24,35,23,38)(12,26,25,32,27,39,20,17)(15,29,30,37,31,36,34,22)

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725385, #11725387, #11725399, …
maximal normal subgroup of	#11725385, #11725387, #11725399, …
maximal quotient of	#11725306, #11725307, #11725614, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 63);