C₉₇×#1404 #27137362

order	12416
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#1404

Subgroups

	subgroup	G/H
center, socle	C₂×C₉₇	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₄×C₉₇
Frattini, cosocle	#124	C₂²×C₉₇
prime radical	#454×C₉₇	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉₇ ◁ C₂²×C₉₇ ◁ C₂×C₄×C₉₇ ◁ #124×C₉₇ ◁ G

112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 113
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 2 generators on 113 points:

(1,2,3,7)(4,13,14,10)(5,9,12,16)(6,15,8,11)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,2)(3,10)(4,13)(5,9)(6,12)(7,11)(8,15)(14,16)

Conjugacy classes & character table

1649 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 136);