C₇×M₆₄ #89412

order	448
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×C₃×#131 (order 384) · Inn(G) = C₂² · Out(G) = C₂³×C₃×C₄
homology	Abelianization (H₁) = C₂×C₇×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×M₆₄

Subgroups

	subgroup	G/H
center	C₇×C₁₆	C₂²
commutator	C₂	C₂×C₇×C₁₆
Frattini, cosocle	C₁₆	C₂²×C₇
socle	C₂×C₇	C₂×C₁₆
prime radical	C₂²×C₇	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₁₆ ◁ G

32 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:

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#11725405, #11725406, #11725407, …
maximal subgroup of	#11725405, #11725406, #11725407, …
maximal normal subgroup of	#11725405, #11725406, #11725407, …
maximal quotient of	C₇×#1397, C₇×C₂²⋊C₃₂, C₇×C₄⋊C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 174);

C7×M64 #89412