C₇×#12777 #13189993

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#12777

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₇
prime radical	C₂³×C₇	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂²×C₇×Q₈ ◁ G

784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5765);