C₁₇×#1868 #14524874

order	2176
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#1868

Subgroups

	subgroup	G/H
center	C₄×C₁₇	#120
commutator	C₂³	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
socle	C₂×C₁₇	#385
prime radical	C₂×#41×C₁₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ G

260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 49
transitive degree	μ_t(G) = 544

A minimal faithful permutation representation — 3 generators on 49 points:

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

Conjugacy classes & character table

544 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16951102, #16951109, #16951136, …
maximal subgroup of	C₁₇×#12962, C₁₇×#12963, C₁₇×#12968, …
maximal normal subgroup of	C₁₇×#12962, C₁₇×#12963, C₁₇×#12968, …
maximal quotient of	C₁₇×#8851, C₁₇×#8852, C₁₇×#8861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 600);