C₂³×#13133296 #28330800

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product	C₂³×#13133296
unambiguous semidirect product	D₅₃⋊(C₂²×#117)

Splittings

Fitting	C₂⁴×C₄×C₅₃ ⋊ C₄
Sylow	C₅₃ ⋊ C₂³×#117

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×#6157
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#6157
Fitting	C₂⁴×C₄×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×D₅₃	C₂²
cosocle	C₂×D₅₃	C₂⁶
hypercenter	C₂⁴×C₄	#6157
hypocenter	C₅₃	C₂³×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

14334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 6 generators on 67 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1116276);

C23×#13133296 #28330800