C₂²×#15690884 #27158697

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×#15690884

Splittings

Sylow

C₉₇ ⋊ C₂²×M₃₂

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#87604
commutator	C₂×C₉₇	C₂³×C₈
Frattini	C₄	C₂³×#87604
Fitting	C₂³×C₄×C₉₇	C₄
socle	C₂³×C₉₇	C₂×C₈
prime radical	C₂⁴×C₉₇	C₈
cosocle	#11715895	C₂⁴
hypercenter	C₂³×C₄	#87604
hypocenter	C₉₇	C₂²×M₃₂
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 6208

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21471);

C22×#15690884 #27158697