C₂×#383×D₅₃ #27542368

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#383×D₅₃

Splittings

Fitting	C₂×C₅₃×#383 ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#383

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#383	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁴×D₅₃	C₂³
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#383	D₅₃
hypocenter	C₅₃	C₂²×#383
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#383

3678 conjugacy classes of subgroups; only the 1219 normal subgroups are shown.

normal subgroups only

This group has 3678 conjugacy classes of subgroups — too many to draw. Only its 1219 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) = 75
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 75 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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(61,65)(64,67)(68,70,71,75)(69,73,72,74);(60,62,63,66)(61,64,65,67)(69,72)(73,74);(56,57,58,59)(60,61)(62,64)(63,65)(66,67)(68,69)(70,74)(71,72)(73,75);(54,55)

Conjugacy classes & character table

1568 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327844);