C₇×#12034 #13189250

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#12034

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#107
prime radical	C₂³×C₇	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 55
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23)(24,25)(26,35)(27,32)(28,42)(29,44)(30,34)(31,38)(33,36)(37,48)(39,54)(40,50)(41,47)(43,55)(45,52)(46,53)(49,51);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23)(24,26,36,51,29,39,43,53,30,40,41,52,27,37,28,38)(25,31,42,48,32,45,47,50,34,46,55,54,44,49,33,35);(10,16)(13,19)(17,22)(20,23)(24,27,30,29)(25,32,34,44)(26,37,40,39)(28,41,43,36)(31,45,46,49)(33,42,47,55)(35,48,50,54)(38,52,53,51)

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5022);