D₄×#13095519 #27158439

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₄×#13095519

Splittings

Sylow

C₉₇ ⋊ C₂×C₈×D₄

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#87604
commutator	C₂×C₉₇	C₂³×C₈
Frattini	C₂²	C₂³×#87604
Fitting	C₄×D₄×C₉₇	C₄
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂×D₄×D₉₇	C₄
cosocle	C₂×Dic₉₇	C₂⁴
hypercenter	C₄×D₄	#87604
hypocenter	C₉₇	C₂×C₈×D₄
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 109 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21213);

D4×#13095519 #27158439