C₁₇×#2115 #14525121

order	2176
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#2115

Subgroups

	subgroup	G/H
center	C₈×C₁₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₁₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₇
socle	C₂×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₇	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ G

220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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)(18,19,21,26,22,27,33,41,23,25,34,38,35,39,46,48)(20,28,30,40,31,42,44,49,29,24,32,36,43,37,45,47);(18,20)(19,24)(21,32)(22,31)(23,29)(25,28)(26,40)(27,37)(30,34)(33,45)(35,43)(36,38)(39,42)(41,49)(44,46)(47,48);(20,29)(24,28)(30,32)(31,43)(36,40)(37,42)(44,45)(47,49)

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16921330, #16921333, #16943624, …
maximal subgroup of	C₁₇×#7207, C₁₇×#7208, C₁₇×#11483, …
maximal normal subgroup of	C₁₇×#7207, C₁₇×#7208, C₁₇×#11483, …
maximal quotient of	C₁₇×#11419, C₁₇×#11441, C₁₇×#11461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 847);