C₇×#7383 #13184599

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7383

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	D₁₆
commutator	C₁₆	C₂×C₇×C₈
Frattini, cosocle	C₄×C₁₆	C₂²×C₇
socle	C₂²×C₇	#357
prime radical	C₂×C₇×D₁₆	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

138 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 47
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 2 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(16,17)(18,22)(19,31)(20,34)(21,26)(23,27)(24,28)(25,32)(29,41)(30,37)(33,47)(35,43)(36,42)(38,46)(39,44)(40,45);(17,22)(18,27)(19,28)(20,32)(23,24)(25,41)(26,37)(29,46)(30,44)(31,35)(33,45)(34,36)(38,47)(39,40)(42,43)

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, 371);