C₂³×#13133179 #28290656

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂³×#13133179

Splittings

Fitting	C₂³×C₄⋊C₄×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂³×C₄×D₄

Subgroups

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁴×D₂₁₂	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

19998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 6784

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

(1,2)(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)(60,62,63,61);(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)(65,67);(60,61,63,62)(64,65)(66,67);(58,59);(56,57);(54,55)

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1076132);

C23×#13133179 #28290656