C₂×#24915727 #35610815

order	21248
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#24915727

Splittings

Sylow

C₈₃ ⋊ C₂×#2842

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₈₃
commutator, cosocle	C₂³×C₈₃	C₂⁵
Frattini	C₂³	C₂⁴×D₈₃
Fitting	C₂×#379×C₈₃	C₂
socle	C₂⁴×C₈₃	C₂⁴
prime radical	C₂⁶×C₈₃	C₂²
hypercenter	C₂×#379	D₈₃
hypocenter	C₈₃	C₂×#2842
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#379

3294 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 667 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) = 109
transitive degree	μ_t(G) = 5312

A minimal faithful permutation representation — 5 generators on 109 points:

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

Conjugacy classes & character table

2348 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 375979);