C₂³×Q₈×#6157 #28330802

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

Aliases

direct product	C₂³×Q₈×#6157
unambiguous semidirect product	D₅₃⋊(C₂²×C₄×Q₈)

Splittings

Fitting	C₂³×Q₈×C₅₃ ⋊ C₄
Sylow	C₅₃ ⋊ C₂³×C₄×Q₈

Subgroups

	subgroup	G/H
center	C₂⁴	C₂²×#6157
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#6157
Fitting	C₂³×Q₈×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁴×D₅₃	C₂³
cosocle	C₂×D₅₃	C₂⁶
hypercenter	C₂³×Q₈	#6157
hypocenter	C₅₃	C₂³×C₄×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

12446 conjugacy classes of subgroups; only the 6264 normal subgroups are shown.

normal subgroups only

This group has 12446 conjugacy classes of subgroups — too many to draw. Only its 6264 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) = 67
transitive degree	μ_t(G) = 3392

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1116278);