C₇×#12837 #13190053

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12837

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#107
prime radical	C₂⁴×C₇	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

394 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 43
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,16,21,18,22,17,32)(14,23,26,38,28,39,27,42)(15,29,30,36,31,43,24,20)(19,33,34,41,35,40,37,25);(12,14,24,40,17,27,31,41,18,28,30,33,16,26,15,25)(13,19,29,38,21,34,36,39,22,35,43,42,32,37,20,23);(9,11)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5825);