C₂⁴×C₄⋊Dic₅₃ #28290647

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

Aliases

direct product

C₂⁴×C₄⋊Dic₅₃

Splittings

Sylow

C₅₃ ⋊ C₂⁴×C₄⋊C₄

Subgroups

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

19742 conjugacy classes of subgroups; only the 13003 normal subgroups are shown.

normal subgroups only

This group has 19742 conjugacy classes of subgroups — too many to draw. Only its 13003 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) = 69
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 6 generators on 69 points:

(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)(66,67,68,69);(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)(62,63,64,65)(67,69);(60,61);(58,59);(56,57);(54,55)

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1076123);