C₂×#20261570 #27156543

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#20261570

Splittings

Fitting	C₂×H(5,𝔽₂)×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ C₂×#585

Subgroups

	subgroup	G/H
center	C₂²	C₂⁴×D₉₇
commutator, cosocle	C₂×C₉₇	C₂⁶
Frattini	C₂	C₂⁵×D₉₇
Fitting	C₂×H(5,𝔽₂)×C₉₇	C₂
socle	C₂²×C₉₇	C₂⁵
hypercenter	C₂×H(5,𝔽₂)	D₉₇
hypocenter	C₉₇	C₂×#585
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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) = 115
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 6 generators on 115 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(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)(100,103,104,111)(101,107,106,113)(102,110,109,115)(105,112,108,114);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(101,106)(105,108)(107,113)(112,114);(102,109)(105,108)(110,115)(112,114);(100,102)(101,105)(103,110)(104,109)(106,108)(107,112)(111,115)(113,114);(100,101)(102,108)(103,107)(104,106)(105,109)(110,114)(111,113)(112,115);(98,99)

Conjugacy classes & character table

1700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 19317);