C₉₇×#1749 #27137707

order	12416
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₉₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉₇×#1749

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₉₇	C₂³
commutator	C₂²	C₂×C₄²×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
socle	C₂³×C₉₇	C₂²×C₄
prime radical	C₂⁴×C₉₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉₇ ◁ G

412 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 6208

A minimal faithful permutation representation — 3 generators on 113 points:

(1,2,4,3)(5,6)(7,8)(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);(6,8)(9,10,11,14,12,13,15,16);(1,3,4,2)(10,13)(14,16)

Conjugacy classes & character table

5432 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉₇×#7021, C₉₇×#7033, C₉₇×#7036, …
maximal normal subgroup of	C₉₇×#7021, C₉₇×#7033, C₉₇×#7036, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 481);