#27143495

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #2151

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	#405×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂×C₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#405	D₉₇
hypocenter	C₉₇	#2151
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 133 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(104,112)(108,118)(113,124)(114,125)(119,128)(120,129)(126,132)(130,133);(98,99,100,101)(102,104)(103,108)(105,113)(106,114)(107,112)(109,119)(110,120)(111,118)(115,126)(116,124)(117,125)(121,130)(122,128)(123,129)(127,132)(131,133);(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(99,101)(102,103,105,109,106,110,115,121,107,111,116,122,117,123,127,131)(104,108,113,119,114,120,126,130,112,118,124,128,125,129,132,133)

Polycyclic presentation:

2^7,97|a^2=e,b^2=d,e^2=f,f^2=g,b^a=bd,h^a=h^96,c^b=cg

Conjugacy classes & character table

1976 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 6269);