C₂×#20277670 #27543448

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#20277670

Splittings

Sylow

C₅₃ ⋊ C₂×#2355

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂⁴
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂×#2355
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄

2638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 5 generators on 75 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)(68,70,71,75)(69,73,72,74);(60,63,64,62)(61,65,66,67);(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)(60,62,64,63)(61,65,66,67)(68,69,71,72)(70,74,75,73);(56,57,58,59)(60,61)(62,65)(63,67)(64,66)(69,72)(73,74);(54,55)

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 328924);

C2×#20277670 #27543448