C₂×Q₆₄×D₅₃ #27542213

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×Q₆₄×D₅₃

Splittings

Fitting	C₂×C₅₃×Q₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×Q₆₄

Subgroups

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂×C₅₃×Q₆₄	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	C₂²×D₅₃	D₁₆
hypercenter	C₂×Q₆₄	D₅₃
hypocenter	C₅₃	C₂²×Q₆₄
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 119
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 119 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)(56,58,62,72)(57,63,67,88)(59,74,78,108)(60,80,82,114)(61,84,81,71)(64,90,92,117)(65,94,95,119)(66,96,83,87)(68,97,99,73)(69,89,85,76)(70,93,102,75)(77,112,110,107)(79,86,113,115)(91,118,104,101)(98,105,103,111)(100,109,106,116);(56,57,62,67)(58,68,72,99)(59,73,78,97)(60,79,82,113)(61,83,81,66)(63,85,88,69)(64,89,92,76)(65,93,95,75)(70,101,102,118)(71,103,84,98)(74,94,108,119)(77,111,110,105)(80,90,114,117)(86,112,115,107)(87,100,96,106)(91,109,104,116);(54,55)

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327689);