C₂³×D₅₃⋊M₁₆ #28330817

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product	C₂³×D₅₃⋊M₁₆
unambiguous semidirect product	D₅₃⋊(C₂³×M₁₆)

Splittings

Sylow

C₅₃ ⋊ C₂⁴×M₁₆

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	C₄
cosocle	Dic₅₃	C₂⁶
hypercenter	C₂⁴×C₄	#6157
hypocenter	C₅₃	C₂⁴×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

13310 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)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,40)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,50)(31,32)(34,49)(35,36)(38,39)(42,43)(44,53)(45,46)(47,48)(51,52);(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, 1116293);

C23×D53⋊M16 #28330817