#27158453

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #2929

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87604
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#123×C₉₇	C₄
prime radical	#20261243	C₂
cosocle	C₂×Dic₉₇	C₂⁴
hypercenter	#123	#87604
hypocenter	C₉₇	#2929
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #123

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 117 points:

(1,2,6,12,22,36,54,76,55,33,49,69,81,53,72,88,91,78,56,38,24,14,5,8,16,28,42,60,82,61,37,19,31,47,67,75,89,95,97,93,83,62,44,27,11,18,30,46,66,85,64,43,23,9,17,29,45,63,84,94,96,90,73,87,65,41,21,32,48,68,80,58,40,26,13,3,7,15,25,39,57,79,92,77,51,71,86,59,35,50,70,74,52,34,20,10,4)(103,108)(107,113)(109,115)(114,117);(2,5,4,3)(6,11,10,9)(7,8,14,13)(12,21,20,19)(15,18,24,23)(16,27,26,17)(22,35,34,33)(25,32,38,37)(28,41,40,31)(29,30,44,43)(36,53,52,51)(39,50,56,55)(42,59,58,49)(45,48,62,61)(46,65,64,47)(54,75,74,73)(57,72,78,77)(60,81,80,71)(63,70,83,76)(66,86,85,69)(67,68,87,82)(79,89,91,90)(84,88,93,92)(94,95,97,96)(102,104,105,110,106,111,112,116)(103,107,109,114,108,113,115,117);(2,4)(3,5)(6,10)(7,14)(8,13)(9,11)(12,20)(15,24)(16,26)(17,27)(18,23)(19,21)(22,34)(25,38)(28,40)(29,44)(30,43)(31,41)(32,37)(33,35)(36,52)(39,56)(42,58)(45,62)(46,64)(47,65)(48,61)(49,59)(50,55)(51,53)(54,74)(57,78)(60,80)(63,83)(66,85)(67,87)(68,82)(69,86)(70,76)(71,81)(72,77)(73,75)(79,91)(84,93)(88,92)(89,90)(94,97)(95,96)(99,101);(2,3,4,5)(6,9,10,11)(7,13,14,8)(12,19,20,21)(15,23,24,18)(16,17,26,27)(22,33,34,35)(25,37,38,32)(28,31,40,41)(29,43,44,30)(36,51,52,53)(39,55,56,50)(42,49,58,59)(45,61,62,48)(46,47,64,65)(54,73,74,75)(57,77,78,72)(60,71,80,81)(63,76,83,70)(66,69,85,86)(67,82,87,68)(79,90,91,89)(84,92,93,88)(94,96,97,95)(98,99)(100,101)(102,103,105,109,106,108,112,115)(104,107,110,114,111,113,116,117)

Polycyclic presentation:

2^7,97|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^a=h^75,h^b=h^96,h^c=h^22,h^f=h^96

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21227);