C₂×(C₂₉⋊C₇)⋊D₂₉ #37555441

order	23548
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×(C₂₉⋊C₇)⋊D₂₉
unambiguous semidirect product	(C₂×C₂₉⋊C₇)⋊D₂₉

Splittings

Fitting, socle	C₂×C₂₉² ⋊ C₂×C₇
derived, Sylow	C₂₉² ⋊ C₂²×C₇
Hall	C₂₉×C₂₉⋊C₇ ⋊ C₂²
Hall	C₂×C₂₉⋊D₂₉ ⋊ C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	(C₂₉⋊C₇)⋊D₂₉
commutator, cosocle, hypocenter	C₂₉²	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₂₉²	C₂×C₇
LCS	G ▷ C₂₉²
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 406

A minimal faithful permutation representation — 2 generators on 60 points:

(2,6,7,21,14,22,28,3,4,11,13,20,26,29)(5,18,19,23,24,15,16,9,10,27,12,25,8,17)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58);(1,2,5,10,20,11,22,13,25,29,27,16,15,6,12,23,4,8,17,19,28,24,21,26,7,14,18,9,3)(30,31,33,35,37,39,41,43,45,47,49,51,53,55,57,58,56,54,52,50,48,46,44,42,40,38,36,34,32)(59,60)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23548, 78);

C2×(C29⋊C7)⋊D29 #37555441